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COPYRIGHT DEPOSm 



ELEMENTS 
/ 



OF 






TRIGONOMETRY, 



PLANE AND SPHERICAL : 



WITH APPLICATIONS TO 



HEIGHTS AND DISTANCES, THE AREAS OF 
POLYGONS, 

SURVEYING, NAVIGATION, 

AND THE SOLUTION OF 

ASTRONOMICAL PROBLEMS. 



JAMES B. DODD, A. M., 

LATE MOBEISON PEOFESSOE OF MATHEMATICS AND KATITEAL PHILOSOPHY IK 
TEANSYLYANIA UNIYEESITY. 




NEW YOKK : 



PRATT, OAKLEY & CO., 

21 MUREAT STREET. 

1858. 



Entered, according to Act of Congress, in the year 1 
By JAMES B. DODD, 
In the Clerk's Office of the District Court of Kentucky. 



WILLIAM DENY8E, 

Stereotyper and Electbotyper, 
183 William Street, New York. 



■■■■■ 



PREFACE. 



This work is designed to follow the author's Treatise 
on the Elements of Geometry and Mensuration, in a 
regular course of mathematical studies. 

It carries the subject of Trigonometry and its Appli- 
cations to the full extent of what has appeared to the 
author to be either necessary or desirable in a scheme 
of liberal education, which should embrace the various 
departments of science in due proportion, and have 
respect to the time within which it is expected to be 
accomplished. 

Great pains have been taken to arrange the several 
divisions of the general subject in their natural order, 
and to render their entire exposition as exact and 
perspicuous as possible; to adapt the whole to the 
comprehension of the student and the convenience of the 
teacher, under the conviction that the utility and 
ultimate success of the work, as a manual of instruction 
and exercise in schools, could be secured only by the 
attainment of both these objects. 

The numerous applicatory Problems which have been 
introduced, in connection with special subjects expounded, 



IV PREFACE. 

and in a regular series at the close of the work, 
will suggest to the student the indispensable utility 
of Trigonometry and its dependent sciences ; and should 
stimulate him to a thorough mastery of the principles and 
methods concerned in their solution. 

The numerical solution of all these Problems would 
occupy more of the student's time than could be advan- 
tageously employed in this manner. When he has become 
familiar with the different methods of computation required 
in any Problem, he may give only a theoretical solution, 
that is, describe the process of solution, without the 
numerical operations. This will secure to him all the 
improvement to be derived from the study of numerous 
problems, with the saving of much unproductive labor. 

The astronomical Problems are designed to exemplify 
some of the uses of Spherical Trigonometry, in order 
that this branch of the science might not be presented 
as a mere theory, unconnected with any of the practical 
utilities of life. A more extensive application of this 
part of the subject is appropriate to Practical Astronomy, 
and inconsistent with the objects of a purely mathemati- 
cal treatise. 

The author desires that this work be appreciated in 
view of its avowed design. It aims at such a course of 
study, on the subjects embraced, as is required for the 
purposes of a liberal education. This general design does 
not include every thing that might be useful to the 
practical Surveyor, Navigator, or Engineer; and it is 
not best to interrupt the consecutive study of the general 



PllEFACE. V 

course of science, which, as a means of intellectual 
culture, is alike important to all, by a multiplicity of 
practical dictations that can be interesting or useful 
to but very few. We have given such descriptions of 
mathematical instruments as seemed necessary to render 
their use intelligible, but have left something to be 
learned in practice, where it can be most availably 
learned, from an inspection of the instruments them- 
selves. "We have established the principles on which 
practical operations depend, and have made such appli- 
cations as properly come within the province of the 
mathematical teacher in the school, without seeking to 
specify every expedient that might be adopted, in Sur- 
veying, for example, in the field. There are works 
already before the public which are particularly adapted 
to field operations : these works may be recommended 
to the student destined for professional Surveying or 
Engineering, after he shall have completed a general 
course of mathematical studies. 

It only remains to be said, that the Mathematical Tables 
appended have been printed in distinct and convenient 
form, in moderate compass, and in exact agreement with 
the most reputable of those already extant. 



Transylvania University. 
May U\ 185a 



CONTENTS. 



BOOK I. 

PAGB 

Properties of Logarithms l 

Definition of Logarithms 1 

Characteristics and Mantissas of Logarithms 2 

Logarithm of a Decimal Fraction 2 

Description of the Table of Logarithms 2 

Application of Logarithms 7 

Logarithmic Multiplication, Division, &c 7 

Complements of Logarithms 7 

Accuracy of Logarithmic Tables 10 



BOOK II. 

Plane Trigonometry, and its more immediate applications 11 

Definition of Plane Trigonometry 11 

Degrees and the Measure of Angles 12 

Complements and Supplements of Arcs or Angles 12 

Trigonometrical Lines 13 

Positive and Negative Sines, Tangents, &c 14 

Negative Arcs 15 

Sines, Tangents, &c, of Supplemental Arcs 16 

Sines, Tangents, &c, of Negative Arcs 17 

Mutations of Sines, Tangents, &c, and their Values for Certain Arcs 17 

Sines, Tangents, &c, of Arcs exceeding a Circumference 19 

Proportions and Formulas 20 

Radius regarded as Unity 21 

Sine and Cosine of the Sum and Difference of two Arcs 23 

Sine and Cosine of a Double Arc 24 

Sine and Cosine of Half a given Arc 24 

vii 



Vlll CONTENTS. 

PAGE 

Products of Sines and Cosines 25 

Various Relations of the Sine and Cosine of two Arcs 26 

Tangent and Cotangent of the Sum and Difference of two Arcs 27 

Tangents and Cotangents of Multiple Arcs 28 

Natural Sines, Tagents, &c... 29 

Definitions of Natural Sine, Tangent, &c 29 

Computation of Natural Sines and Cosines 29 

Computation of Natural Tangents, Cotangents, Secants, and Cosecants 30 

Description of the Table of Natural Sines and Cosines 30 

Logarithmic Sines, Tangents, &c. 31 

Definitions of Logarithmic Sines, Tangents, &c 31 

Computation of Logarithmic Sines, Tangents, &c. 32 

Description of the Table of Logarithmic Sines, Tangents, &c 32 

Theorems in Plane Trigonometry 35 

Solutions of Plane Triangles ;, 40 

General Rule for the Solution of Right-angled Triangles 41 

Solutions of any Plane Triangles 47 

Graphic or Instrumental Solutions 55 

Heights and Distances 61 

Vertical and Horizontal Lines, Angles, &c 61 

Instrumental Measurement of Angles 62 

Use of Vernier 62 

Problems in Heights and Distances 64 

Areas of Polygons - 70 



BOOK III. 

Surveying 78 

Definition of Surveying, and of a Level Surface 78 

Deviation of a Level Surface from a Plane 78 

Plane Surveying 79 

Measurement of Distances in Surveying 80 

Expression of Areas in Surveying 80 

Geographical and Magnetic Meridians 81 

Bearing of a Line 81. 

Measurement of Bearings, and Description of the Compass 82 

Forward and Back Sights 84 

Difference of Latitude and Departure 84 



CONTENTS. IX 

PAGE 

Computation of Latitudes and Departures 85 

Description of the Traverse Table „ 86 

Field Notes of a Survey 88 

Plotting a Survey 90 

Balancing the Latitudes and Departures 91 

Area of the Survey 96 

The Area computed from the Latitudes and Departures 96 

Graphic Method of computing the Area 99 

The Area found from the Interior Angles and the Sides 103 

Surveying by Offsets, in Particular Cases 106 

Division of Land 108 

Problems in the Division of Triangles and other Polygons 108 

Variation of the Magnetic Needle 121 

Surveys of the Public Lands of the United States 128 

Leveling, with a Description of the Leveling Instrument 130 

Topography, with a Description of Topographical Maps. 133 

Geodesic Surveying, with a Description of the Theodolite 135 

Figure and Magnitude of the Earth 142 



LOOK IV. 

Navigation 144 

Navigation Defined — Latitude and Longitude 144 

Ship's Course, with a Description of the Mariner's Compass 145 

Ship's Rate of Sailing, how determined 147 

Plane Sailing 148 

The Ship's Departure on a Uniform Course 148 

Problems in Plane Sailing solved by Plane Triangles 149 

Traverse Sailing, and circumstances affecting its Accuracy 149 

Parallel Sailing " 153 

Parallel Sailing defined, with the Proportion involved. 153 

Problems in Parallel Sailing solved by Plane Triangles 154 

Middle Latitude Sailing 154 

Correction of the Middle Latitude Method. 156 

Mercator's Sailing 160 

Mercator's Sailing defined — Meridional Difference of Latitude 160 

Description of the Table of Meridional Parts 161 

Computation of the Meridional Parts 161 

Formula for the Corrections of the Middle Latitude Method .-. 164 

Uncertainty of the Ship's Place by the Dead Reckoning 165 

1* 



CONTENTS 



BOOK V. 

PAGB 

Spherical Trigonometry, and its more immediate applications.. 167 

Definition of Spherical Trigonometry 167 

Theorems in Spherical Trigonometry 167 

Solution of Spherical Triangles , 177 

Napier's Theorem for the Solution of Right-angled Spherical Triangles 177 

General Rule for Right-angled Spherical Triangles 180 

Solutions of Quadrantal Triangles 183 

Solutions of Oblique-angled Spherical Triangles . .. 184 

Applications of Spherical Trigonometry 193 

Problems on the Terrestrial Sphere „ 194 

Connection of Spherical Trigonometry with Astronomy 195 

Description of the Celestial Sphere 196 

Problems on the Celestial Sphere 199 

Miscellaneous Problems in the Applications of Trigonometry... 205 



ELEMENTS OF TRIGONOMETRY. 



BOOK I. 

PROPERTIES OP LOGARITHMS. 

Logarithms have no connection with the Theory of 
Trigonometry, but are used to facilitate trigonometrical 
calculations. So far only as is necessary for this purpose 
their properties will be here exhibited ; for a fuller exposi- 
tion, the student is referred to the author's Algebra. 



(1.) Logarithms of numbers are the exponents of the 
powers to which some assumed number, called the "base of 
the logarithms, must be raised, to produce the given num- 
bers. 

The common system of Logarithms is based on the number 
10 ; all numbers being regarded, in this system, as powers 
o/lO. 

is the logarithm of 1, because 10° equals 1 ; (Algebra, 48) ; 

1 is the logarithm of 10, because 10 1 equals 10 ; 

2 is the logarithm of 100, because 10 2 equals 100 ; 

3 is the logarithm of 1000, because 10 3 equals 1000, &c. 

4 is the logarithm of what number ? 5 is the logarithm of 
what number ? 6 is the logarithm of what number ? 

The logarithm of a number which is not equal to any 
integral power of the base 10, may be computed by the 
methods explained in Algebra (313) ; and thus a Table of 
Logarithms may be formed, for general use. 



LOGARITHMS. 



Characteristics and Mantissas of Logarithms. 

(2.) The integral part of a Logarithm is called the charac- 
teristic, and the decimal part the mantissa, of the logarithm. 

Thus, by computation, the logarithm of 365 is 2.562293; 
that is, 

10 2 - 562293 =365; 

in which 2 is the characteristic, and .562293 is the mantissa, 
of the logarithm. 

(3.) The characteristic of the logarithm of any number 
greater than %mity, is one less than the number of integral 
figures in the given number. 

Thus the number 365 contains three integral figures, and 
the characteristic of its logarithm, as already seen, is 2. 

Logarithm of a Decimal Fraction. 

(4.) The logarithm of a Decimal Fraction has the same 
mantissa as if the given fraction were an integer / but its 
characteristic is negative, and is one more than the number of 
Os immediately after the point (.) in the given decimal (Alg. 
320). 

Thus the logarithm of .365 is 1.562293 ; 



" " .0365 is 2.562293; 

" " .00365 is 3.562293. 

The sign is set over the characteristics 1, 2, 3, to show 
that only these characteristics are negative ; the mantissa of 
every logarithm is positive. 

TABLE OF LOGARITHMS. 

(5.) Table 1., appended to this work, contains the Loga- 
rithms of all numbers from 1 to 10000, or, rather, the 
mantissas of the logarithms, since the characteristics, being 
always known (3 and 4), are omitted for numbers above 
100. These mantissas are given to six places of decimals. 

On the first page of the Table, in the columns marked 1ST., 
at the top, are the natural numbers from 1 to 100, with the 



BOOK I. 3 

logarithm of each immediately on the right, in the next 
column. 

On the second and following pages of the Table, in columns 
!N~., are the natural numbers from 100 to 999, with the man- 
tissa of the logarithm of each immediately on the right, in 
the next column — vacant places in the latter requiring to be 
supplied with the two initial figures which stand next above 
the vacant place. 

Thus the mantissa of the logarithm of 138 is .139879. By 
prefixing the characteristic 2 (3), we find 

The logarithm of 138=2.139879. 

When the natural number consists of four figures , find 
its three left-hand figures in column ]ST., and the other at the 
head of one of the other columns. The mantissa of its loga- 
rithm will consist of the four figures in the latter column 
standing opposite the three left-hand figures of the given 
number, together with the two corresponding initial figures 
in the column under 0. 

Thus the mantissa of the logarithm of 1763 is .246252. 
By prefixing the characteristic 3 (3), we find 

The logarithm of 1763=3.246252. 

Points or dots occurring among the logarithms in the Table, 
denote that such places must be supplied with 0s, and that 
from thence the two initial figures must be taken, in column 
0, from the line next below. 

Thus, the mantissa of the logarithm of 1863 is .270213 ; 
the mantissa of the logarithm of 1S64 is .270446. 

The numbers in columns D, on the right of the Table — : 
with 0s prefixed to each, to make six decimal places — express 
the increase in the logarithm for an increase- of a unit in the 
corresponding natural numbers, being the differences between 
consecutive mantissas on the same horizontal line. These 
numbers are used when it is required 



4 LOGARITHMS. 

(6.) To find from the Table the Logarithm of a Number 
containing more than four Figures. 

1. Find the mantissa corresponding to the four left-hand 
figures of the given number, in the manner already 
explained. 

2. Multiply the number in column D, in line with this 
mantissa, by the remaining figures of the given number, and 
reject from the right of the product as many figures as there 
are in the multiplier. 

3. Add the other part of the product to the mantissa 
already found (as if both were integers) — increasing the right- 
hand figure added by a unit, when the adjacent rejected 
figure is 5 or more. 

4. To the mantissa thus obtained prefix the proper char- 
acteristic (3 or 4). 

EXAMPLE. 

To find the logarithm of 162.731. 

The mantissa corresponding to 1627, in the Table, is 

.211388. 

On the same horizontal line, in column D, we find 267, 
which we multiply by 31, the remaining figures of the given 
number. 

267X31=8277. 

Eejecting two figures from the right of this product — there 
being two figures in the multiplier 31 — we have 83 to add 
to the mantissa already found 

.211388 
83 



,211471 



The characteristic to be prefixed is 2, since the given 
number contains three integral figures (3) ; so that 

The logarithm of 162.731 is 2.211471. 



BOOK I 



EXERCISES. 



1. Find the logarithm of 2200 Ans. 3.342423. 

2. Find the logarithm of 24.36 Ans. 1.386677. 

3. Find the logarithm of 2.698 Ans. 0.431042. 

4. Find the logarithm of 3585.9 Ans. 3.554598. 

5. Find the logarithm of 42.6634 Ans. 1.630056. 

6. Find the logarithm of 331.957 Ans. 2.521082. 

7. Find the logarithm of 2519.38 Ans. 3.401294. 

8. Find the logarithm of .538329 Ans. 1.731047. 

9. Find the logarithm of .087346 Ans. 2.941243. 

10. Find the logarithm of .007389 Ans. 3.868586. 



(7.) To find from the Table the natural Number for any 
given Logarithm. 

1. Find the two initial figures of the given mantissa in the 
column under 0, and the other four, if possible, in that or 
another column ; then in column !N"., opposite to the latter 
figures, will be found the three left-hand figures, and at the 
top of the page the other figure, of the required number. 

2. When the exact mantissa is not contained in the Table, 
find the four figures corresponding to the next less mantissa 
in the table ; subtract this mantissa from the given one ; 
divide the remainder by the number in line in column D (as 
if both these numbers were integers) ; and annex the quotient, 
with the 0s to be prefixed to it, if any, to the four figures 
already found. 

3. The natural number thus obtained must, by a decimal 
point, be made to conform to the characteristic of the given 
logarithm (3 and 4). 

EXAMPLE. 

To find the number whose logarithm is 2.491090. 

The exact mantissa of this logarithm is not contained in 
the Table ; the next less mantissa in the Table will be found 



LOGARITHMS. 



to be .491081, the four figures corresponding to which, in 
column K.j and at the top, are 3098. 

491090 
491081 



140) 9.00 (.06' 

We subtract the mantissa found in the table from the given 
one, and divide the remainder 9 by 140 found in column D. 
This division, performed decimally ', gives the quotient .06'. 
Annexing this quotient to the four figures already found 
from the table, and making three integral figures, for the 
given characteristic 2 (3), we find 

309.806 lor the number whose logarithm is 2.491090. 



EXEKCISES. 

1. Find the number whose logarithm is 1.240050. 

Ans. 17.38. 

2. Find the number whose logarithm is 2.431203. 



3. Find the number whose logarithm is 3.503780 



Ans. 269.9. 

3780. 

Ans. 3189.91. 



4. Find the number whose logarithm is 0.138934. 

Ans. 1.377. 

5. Find the number whose logarithm is 1.368730. 

Ans. .233738. 

6. Find the number whose logarithm is 2.448375. 

Ans. .028078. 

7. Find the number whose logarithm is 3.538630. 

Ans. .003456. 

8. Find the number whose logarithm is .843970. 

Ans. 6.98184. 

9. Find the number whose logarithm is 1.867372. 

Ans. .736837. 
10. Find the number whose logarithm is .003985. 



11. Find the number whose logarithm is 3.723460. 



Ans. 1.00921. 
3460. 

Ans. .005290. 



BOOK I. 



APPLICATION OF LOGARITHMS. 

(8.) Logarithms are used to facilitate tedious operations in 
the multiplication, division, involution, and evolution of 
numbers. These applications of them result from their 
exponential character (1). 

1. To Multiply two or more Numbers together. . 

Add together the logarithms of the given numbers ; the 
sum will be the logarithm of the product of those numbers. 
Find the natural number corresponding to this logarithm (7), 
and it will be the required product (Alg. ,308). 

2. To Divide one given Number by another. 

Subtract the logarithm of the divisor from the logarithm 
of the dividend ; the remainder will be the logarithm of the 
quotient. Find the natural number corresponding to this 
logarithm (7), and it will be the required quotient (Alg. 309). 

3. To raise any Power of a given Number. 

Multiply the logarithm of the given number by the 
exponent of the required power; the product will be the 
logarithm of the required power. Then find the natural 
number corresponding to this logarithm (7) (Alg. 310). 

4. To extract any Root of a given Number. 

Divide the logarithm of the given number by the integer 
corresponding to the required root ; the quotient will be the 
logarithm of the required root. Then find the natural 
number corresponding to this logarithm (7) (Alg. 310). 

COMPLEMENTS OF LOGARITHMS. 

(9.) The complement of a logarithm is wkat remains when 
the logarithm is subtracted from 10. This complement may 
be added in calculations, in which the logarithm itself should 
be subtracted, if 10 be rejected from the sum (Alg. 330). 



8 LOGAKITHMS. 

The complement of a logarithm is most readily obtained 
by subtracting each figure of the logarithm from 9, except 
the last significant figure on the right, which must be sub- 
tracted from 10. 

The complements of logarithms are used to abbreviate a 
calculation in which multiplication and division are both 
concerned. 

EXAMPLES EST LOGARITHMIC CALCULATION. 

To find, logarithmically, a fourth proportional to the 
numbers 

20, 40 and 60. 

The second and third numbers must be multiplied together, 
and their product divided by the first number (Alg. 150). 

Logarithm of 20 .... 1.301030 

" of 40 ... . 1.602060 

« of 60 ... . 1.778151 

120 ... . 2.079181 

Add together the logarithms of 40 and 60 (8...1) ; the sum 
is 3.380211 ; from this sum subtract the logarithm of 20 ; the 
remainder 2.079181 is the logarithm of the product of 40 X 
60 divided by 20 (8.. .2). The corresponding natural number 
is 120, which is therefore the required fourth proportional. 

The calculation will be abbreviated by adding the comple- 
ment of the logarithm of the divisor 20 to the logarithms of 
the factors 40 and 60 of the dividend, and rejecting 10 from 
the sum (9). 

Comp. of log. of 20 ... . 8.698970 

log. of 40 ... . 1.602060 

log. of 60 ... . 1.778151 

120 ... . 2.079181 

The result is the same as before. 

The use of logarithmic complements is most important 
when a number, or the product of two or more numbers, is to 
be divided by the product of two or more numbers. Thus 



BOOK I. 9 



EXAMPLE II. 

To divide the product of 435 and 684 by the product of 
583 and 760, by means of logarithms. 

Comp. of log. of 583 ... . 7.234331 

comp. of log. of 760 ... . 7.119186 

log. of 435 ... . 2.638489 

log. of 684 ... . 2.835056 

Quotient .671524 .... 1.827062 

Add the complements of the logarithms of the two factors 
of the divisor, and the logarithms of the two factors of the 
dividend into one sum, and reject 20 from that sum, because 
two complements are used. 20 from 19 gives negative 1, 
and this negative characteristic makes the corresponding 
natural number a decimal fraction (4). 

example in. 

To extract the Jifth root of .03795. 

The logarithm of the given number is 2.579212 ; the 
characteristic 2 being negative (4), and the mantissa, or 
decimal part of the logarithm, positive, as is always the case 
with the mantissa. 

This logarithm must be divided by 5, to find the logarithm 
of the fifth root of the given number (8... 4). To find an 
exact integral characteristic for the logarithm sought, the 
negative characteristic 2 must be expressed by its equivalent 

2-3+3=54-3. 

(5 +3)-:- 5 gives 1 with remainder 3. This 3 is to be pre- 
fixed to the .5 ; so that the given logarithm divided by 5 gives 

1.715842. 

The natural number corresponding to this logarithm is 
0.519807, which is therefore the required root. 

The preceding method of substituting an equivalent 



10 LOGARITHMS. 

characteristic must be adopted whenever a negative charac- 
teristic is not a multiple of the nnmber by which it is to be 
divided. 

Accuracy of Logarithmic Tables. 

(10.) The mantissas of logarithms, as given in Tables, are 
but approximate decimals, and are therefore more or less 
accurate according to the nnmber of figures they contain. It 
may therefore be observed that, 

" The number of places of figures which may be obtained 
in a result derived from any table of logarithms, is the same 
as the number of decimals to which the logarithms are 
carried. But towards the end of the table, the last place 
thus obtained cannot always be depended upon within a 
unit." " For most practical purposes out of astronomy, and 
for very many of the details of calculation connected with 
the latter science, five places are amply sufficient. " Six 
figures in the mantissas of the logarithms, as in our Table, 
give still more accurate results ; this being the Table which 
is now commonly adopted for educational purposes, in this 
country. . 

Stanley's Mathematical Tables are a very valuable collec- 
tion, in which the mantissas of logarithms are extended to 
seven figures. 



BOOK II. 



PLANE TRIGONOMETRY 

AND ITS MOEE IMMEDIATE APPLICATIONS. 



(11.) Plane Trigonometry is that branch of Mathematics 
which treats of the relations among the sides and angles of 
plane triangles. 

The three sides and three angles of a triangle are called 
the six parts of the triangle. When any three of these parts 
of a plane triangle, except the three angles, are given, the 
other three may be computed by Plane Trigonometry. 

Degrees and the Measwre of Angles. 

(12.) In trigonometrical calculations, the circumference of 
a Circle is supposed to be divided into 360 equal parts called 
degrees, each degree into 60 equal parts called minutes, each 
minute into 60 equal parts called seconds. 

Degrees, minutes, and seconds are denoted thus : 

45° 30' 25", 45 deg. 30 mm, 25 sec. 

(13.) An Angle is measured by an arc of a circle described 
from its vertex as a centre, and intercepted between the sides 
of the angle (Geometry, 260). 

Thus, C being the centre of the 
circle, the angle ACB is measured 
by the arc AB. If this arc contains 
40°, for example, the angle ACB 
is called an angle of 40°. 

A right angle, as ACD, is an 
angle of 90°, since it is measured 
by a quadrant, AD, or fourth part 
of the circumference ; and two right angles together contain 
J80°. 





12 PLANE TRIGONOMETRY. 

Complements and Supplements of Arcs or Angles. 

(14.) The complement of an arc, or angle, is what remains 
after subtracting that arc, or angle, from 90° ; the supplement 
of an arc, or angle, is what remains after subtracting that arc, 
or angle, from 180°. 

Let the arc AD be a quadrant, 
90°, and ADF a semi-circumfer- 
ence, 180° ; then BD is the com- 
plement of AB, and BF is the sup- 
plement of AB. 

Let A represent any arc or 
angle; then 

90°— A is the complement of A ; 
180° —A is the supplement of A. 

Since an arc and its supplement together contain 180°, we 
have also 

The supplement of (90 o -A)=180°-(90 o -A)=90°+A. 

When an arc exceeds 90° its complement is negative; 
when it exceeds 180° its supplement is negative. 
Thus the complement of 100° is 

90 o -100°=-10°; 

the supplement of 200° is 

180°-200°=-20°. 

(15.) The two acute angles of every right-angled triangle 
are complements of each other (Geom. 40) ; and any angle of 
a triangle is the supplement of the sum of the other two 
(Geom. 38). 

The trigonometrical relations of angles are established 
through the medium of certain straight lines connected with 
the arcs which measure the angles, and thence called 



BOOK II. 



13 



Trigonometrical Lines. 

(16.) The Sine of an arc is a straight line drawn from one 
extremity of the arc perpendicular to a diameter drawn to 
the other extremity. 



Thus BG- is the sine of the arc 
AB. 

An arc and its supplement have the 
same sine. For the arc BDF is the 
supplement of AB, and, according to 
the definition, BG is also the sine of 
this supplement. 




2. The Tangent of an arc is a straight line drawn from one 
extremity of the arc, perpendicular to a radius at that 
extremity, and limited by the prolongation of a straight line 
drawn through the centre and the other extremity. 

Thus AK is the tangent of the arc AB. 

3. The Secant of an arc is that part of a straight line drawn 
through the centre of the circle and one extremity of the arc 
which is included between the centre and a tangent drawn 
from the other extremity. 

Thus CK is the secant of the arc AB. 

4. The Cosine, cotangent, or cosecant of an arc, is the sine, 
tangent, or secant, respectively, of the complement of that arc. 

Thus the arc BD is the complement of AB, and BI is the 
sine of BD (16) ; then BI is the cosine of the arc AB. Also 
DO is the cotangent, and CO the cosecant, of AB. 

5. The cosine of an arc is always equal to that part of the 
radius which is intercepted between the centre of the circle 
and the foot of the sine. 

Thus BI, the cosine of AB, is equal to CG- ; these straight 
lines being opposite sides of a rectangle. 

6. The sine, or cosine, &c. of an Angle is the sine, or cosine, 
&c. of the arc which measures the angle (13). 



u 



PLANE TRIGONOMETRY. 



Thus BG is the sine, and BI or CG is the cosine, of the 
angle ACB, measured by the arc AB. 

Let A represent any arc or angle ; and let cos A denote the 
cosine of A, cot A the cotangent of A, &c. ; then, from what 
has been said, 

cos A=sine (90°— A) ; 

cot A=tang (90°— A) ; 

cosec A=sec (90° — A). 

The Versed Sine of an arc is that part of the diameter of 
the circle which is intercepted between the arc and its sine ; 
thus AG is the versed sine of the arc AB. Versed sines are 
very seldom referred to in Trigonometry. 



Positive and Negative Sines, Tangents, &c, in different 

Quadrants. 

(IT.) The successive quadrants, of 90° each, on the circum- 
ference of a circle, are denominated the first, second, third, 
and fourth, respectively; and a sine, or tangent, &c, is said 
to be in that quadrant in which its arc terminates. 

Thus, proceeding from the point A, in which the circum- 
ference may be supposed to originate, AD is the first 
quadrant, DG the second, GL the third, LA the fourth. 

Sines, tangents, &c, are distinguished as positive and 
negative on account of the opposite directions which they 
assume for arcs terminating in different quadrants. 

All the trigonometrical lines of the first quadrant, as the 
sine IB, tangent AE, and secant OK, are considered positive, 
and their directions are then made 
the criteria of positive and negative 
lines in the other quadrants. 

taius the sine OF of the arc AF 
terminating in the second quadrant, 
is positive, since it has the same 
direction as the positive sine IB 
from the diameter AG; while the 
sines OH and IK of the arcs AFH 




book ir. 15 

and AFHK terminating in the third and fourth quadrants, 
are negative, being on the opposite side of the diameter AG. 

The tangent AS of the arc AF is negative, since its direction 
from the point A is opposite to that of the positive tangent 
AR. The secant CS of the arc AF is negative ; for while the 
positive secant CR is drawn towards the termination B of its 
arc AB, CS is drawn away from the termination F of its arc 
AF. The tangent AR of the arc AFIT is positive ; the secant 
CR of the same arc is negative, being drawn away from the 
point H. The tangent AS of the arc AFHK is negative ; 
the secant CS is positive, since it is drawn towards the termi- 
nation of its arc AFHK. 

The positive and negative qualities of cosines, cotangents, 
and cosecants in the different quadrants, may be ascertained 
from the consideration of 

Negative Arcs. 

(l§.) When arcs taken in one direction on the circumference 
of a circle are regarded as positive ; those taken from the 
same point, in the opposite direction, must be regarded as 
negative. 

Thus the arcs AB, AF, &c, being considered positive, AX, 
ALH, &c, must be considered negative. 

Now the cosine of the arc AF, terminating in the second 
quadrant, is the sine of the complement of this arc. This 
complement is 

AD— (AD+DF)= -DF (14). 

This complement being negative, suppose it to be the 
negative arc AK. The sine of AK is XI, and is negative, 
because it lies opposite to the positive sine BI. Hence 
cosines in the second quadrant are negative. 

The tangent of the complement AX is AS, and is negative, 
because it lies opposite to the positive tangent AR in the first 
quadrant. Hence cotangents in the second quadrant are 
negative. 

The secant of AX is CS, and is positive, because it is drawn 
from the centre towards the termination X of its arc AX. 
Cosecants in the second quadrant are therefore positive. 

2 



16 



PLANE TRIGONOMETRY. 



In like manner may be ascertained the positive or negative 
quality of a cosine, cotangent, or cosecant in the third or 
fourth quadrant. 

But a more direct method may be adopted for the cosine. 

Thus CI is the cosine of the arcs AB and AFHK (16... 5), 
and is positive — all lines being regarded as positive which 
are perpendicular to the diameter DL, on the right ; CO is 
the cosine of AF and AFII, and is negative, because it lies 
opposite to the positive cosine CI. 

From articles (17) and (IS) we have therefore the following 

(19.) Table of positive and negative trigonometrical lines 
in the different quadrants ; from which it will be seen that, 
with respect to their algebraic signs, these lines may be 
arranged m pairs. 

8d qd. 4th qd. 



+ — 





In 1st qd. 


2&qd. 


Sin. and cosec. 


+ 


+ 


Cos. and sec. 


+ 





Tang, and cot. 


+ 






Sines , Tangents, <&c, of Supplemental Arcs. 

(20.) The sine of an arc is equal to the sine of its supple- 
ment ; and the cosine of an arc is equal to the cosine of its 
supplement, with a contrary algebraic sign. 



Let AB be equal to FG. The 
supplement of the arc AB is BFG, 
equal to AF terminating in the 
second quadrant. The sines BI and 
FO of the arcs AB and AF are 
equal; and their cosines .CI and 
CO are equal, with contrary signs 
(19). 

2. The tangent, or cotangent, of an arc is equal to the 
tangent, or cotangent, of its supplement, with a contrary 
algebraic sign, 







BOOK II. 



17 



The supplemental arcs AB and AF have the equal tangents 
AH and AS, with contrary signs. The complements of AB 
and AF are BD and — DF or AK, which evidently have 
equal tangents (these being the cotangents of AB and AF), 
with contrary signs. 

We may, in a similar manner, compare the trigonometrical 
lines of arcs terminating in the third and fourth quadrants 
with those of their supplements. 

The general result may be expressed as follows : 

sin A=sin (180°- A) ; cos A=-cos (180°- A) ; 
tanA=-tan (180°-A); cot A=-cot (180°- A); 
sec A=— sec (180° —A) ; cosec A=cosec (180°— A). 

Smes, Cosines, Tangents, <&c, of Negative Arcs. 

(21.) Let —A represent the negative arc AK, and A the 
equal positive arc AB. By comparing the sines, tangents, 
and secants of these arcs with those of their complements 
KBD, or AF, and BD, it will be found that 

sin (— A)=— sin A ; cos (— A)=cos A ; 
tan(— A)=— tan A ; cot (— A)=— cot A ; 
sec (— A)=sec A; cosec (— A)=— cosec A. 



Mutations of Sines, Tangents, &c, 
and their Yaluesfor Certain Arcs. 

(22.) "When an Arc increases con- 
tinually from to 360°, its sine, tan- 
gent, &c, undergo infinite mutations ; 
and, at certain points, have values 
which it is important to notice. 

1. Suppose the arc to begin in 
at the point A, and to increase in the 
direction ABDF, &c. At A the sine 
and tangent are evidently 0, while 
the cosine and secant are each equal 
to the radius CA; that is, denoting 
the radius by K, 




13 PLANE TRIGONOMETRY. 

sin or tan 0=0 ; cos or sec 0=K. 



2. The sine of 30° is evidently half 
the chord of 60°, and this chord is 
equal to the radins of the circle 
(Geom. 128); hence 

sin 30°=cos 60°=i E (16...4). 

3. When the arc AB is 45°, the 
triangles GIB and CAK are evidently 
isosceles, so that CI is equal to IB, 
and CA to AE ; hence 




sin 45°=cos 45 == V /JCB 2 (G. 148) 

=EVMAlg.237); 

and tan 45°=cot 45°=E. 



4. The sum of the squares of the sine and cosine is always 
equal to the square of the radius, as in the triangle CIB. 
We have therefore 



sin 2 60°+cos 2 60°=E 2 , and sin 60°= v /E 2 -cos 2 60° ; 

or sin 60°=cos 30°=yE 2 -^E 2 (22...2) 

-yf^=iKV3(Alg.237). 

5. As the arc approaches the point D, the sines approach 
the radius CD ; the cosines diminish towards the centre C, 
and the tangents and secants rapidly increase. The sine of 
AD is CD, the cosine is ; the tangent and secant are infinite, 
because CD produced would never meet AP produced (Geom. 
16...12). 

Hence, adopting the usual symbol of infinity, 

sin 90°=E; cos 90°=0 ; tan or sec 90°=oo . 
Then, cot 0=tan 90°=oo (16...4). 

6. As the arc increases from D towards G, the sines 
diminish, the cosines increase from C on the radius CG, the 
tangents, AT, AS, rapidly diminish towards A, and the 



BOOK II. 19 

secants, CT, CS, approach the radius CA. At G the sine and 
tangent are 0, the cosine is— CG, the secant is CA, and is 
negative, because it is drawn away from the point G. Hence 

sin or tan 180°=0 ; cos 180°=-R ; sec 180°=-R; 
then cot 180°=tan (-90°) = -tan 90° (21)= -oo ; 
and cosec 180°=sec (-90°)= sec 90° (21)= oo . 

7. As the arc increases from G towards L, the sines 
approach the radius CL, the cosines diminish towards C, the 
tangents, AR, AP, and the secants, CR, CP, rapidly increase. 
At L the sine is CL, the cosine is 0, the tangents and secants 
are infinite, because CD produced would never meet AP 
produced. Hence 

sin 270°=— K ; cos 270°=0 ; tan or sec 270°=oo ; 
then cot 270°=tan (-180°)= -tan 180°=0 ; 
and cosec 270°=sec (-180°)=sec 180°= -R. 

8. As the arc increases from L towards A, the sines 
diminish, the cosines increase from C on the radius CA, the 
tangents, AT, AS, rapidly diminish towards A, and the 
secants, CT, CS, approach the radius CA. When the arc has 
increased through the entire circumference, the several 
trigonometrical lines will have passed through all possible 
mutations, and resumed their values at the beginning, that 
is, when the arc was 0. From preceding results we shall 
therefore have 

sin or tan 360°=0 ; cos or sec 360° =R; 
cot or cosec 360°=oo . 

Sines, Tangents, c&c, of Ares Increased by one or more 
Circumferences. 

(23.) If an arc be increased by one or more circumferences, 
it is plain that the arc will still terminate in the same point 
as at first ; and the sine, tangent, &c, of the augmented arc 
will be the same as of the original one. 

Hence it follows that 



20 PLANE TRIGONOMETRY. 

sin A=sin (360°+A)=sin (720°+A), &c. 
cosA=cos(360°+A)=cos (720°+A), &c. 

&C., &C, &Q. 

The sine, or tangent, &c, of an are exceeding a circum- 
ference may therefore always be found in the sine, or tangent, 
&c, of an arc not exceeding a quadrant : 

For let the arc be represented by A. From this arc 
subtract the circumferences which are contained in it ; and 
represent the remainder by B. Then, from what has been 
shown above, 

sin A=sin B. 

If B exceed a semi-circumference, let it be equal to 180°+ 

C; then 

sin B=sin (-C)=-sin (20) and (21). 

If C exceed a quadrant (as it may) let it be equal to 90°+ 
D; then 

-sin C=-sin (90°-D) (20). 

The sine of A will thus be found in the sine of an arc less 
than 90°. Similar methods might be pursued with the other 
trigonometrical lines. 



PROPORTIONS AND FORMULAS. 



(24.) The Proportions subsisting among the trigonomet- 
rical lines of the same arc and the radius of the circle, are 
those of the sides of similar triangles. 

The sine, tangent, cosine, &c. of the 
arc AB are the sides of the triangles CIB, 
CAP, CDF, (DB being the complement 
of AB) ; and these triangles are similar 
to one another (Geom. 218). 

Denoting the arc AB by a, and the 
radius CA, CB, or CD by E, we have the 
following proportions : 




BOOK II. 21 

1. cos a, 01, : sin a, IB, : : K, CA, : tan a, AP ; 

or cos a .tan &=sin a . R. 

2. cos a, CI, : E, OB, : : R, CA, : sec a, OP ; 

or cos a . sec &=R 2 . 

3. sin a, IB, : cos a, CI, : : R, CD, : cot a, DF ; 

or sin a . cot #=eos <z . R. 

4. sin «, IB, : R, CB, : : R, CD, : cosec a, CF ; 

or sin a . cosec #=R 2 . 

5. tan a, AP, : R, CA, : : R, CD, : cot a, DF; 

or tan a . cot a=R 2 . 

6. In the right-angled triangle CIB, we have 

CI 2 -HB 2 =CB 2 ; 
cos 2 &+sin 2 a=R 2 , 



which gives sin a= -y/U 2 — cos 2 a ; 



cos a= -y/R 2 — sin 2 a. 

(25.) Scholium. In the preceding proportions, the trigo* 
nometrical lines are all in the first quadrant. Like propor- 
tions obtain in the other quadrants, since the sine, tangent, 
&c, of an arc in any of those quadrants are equal to those of 
an arc in the first. 

These lines have contrary algebraic signs in different 
quadrants (19) ; but the radius is always a positive term, 
because it always lies towards the arc. In this respect it 
differs from the secant, which sometimes lies away from its 
associated arc. 

JZadius regarded as Unity. 

(26.) The formulas of Trigonometry will be simplified by 
assuming the radius of the circle to be unity, that is, the 
unit of measure of the sines, tangents, &c. 

In the formulas of the preceding Article, let R be made 
one or unity / then 



22 PLANE TRIGONOMETRY. 

1. cos a . tan &=sin a ; 2. cos « . sec a— 1 ; 

sin a . 1 



or 


tan «- 


cos a 




or 


sec <2— > 

cos a 




i a 


cot a= 


cos a; 


4. sin a . 


cosec <x= 


1 


> 




cot #== 


cos a 




or 


cosec <x= 


1 

sin a 




or 


"sin a' . 


? 




5. tan 


a . cot # 


=1, or tan a 


1 

"""cot a 








which 


6. cos 2 
gives sin 


a+sm 2 a— 1, 










0=yi- 


-cos 


2 *, 








cos a—-\/\ 


—sin 


2 a. 







In this connection it should be observed that 

(ft7.) "When a trigonometrical formula does not exhibit the 
radius, and is not homogeneous (Alg. 23), it is because the 
radius is assumed to be unity, and does not appear, in its 
appropriate relations, as a linear quantity. 

Such formula may be adapted to any other value of the 
radius, as H, by substituting the ratio of the sine, or tangent, 
&c, to B for the simple sine, or tangent, &c, since the radius 
K. is thus made the unit of measure. 

Proceeding in this manner with the formula 

cos a . tan a=sin a, 

, cos a . tan a sin a 

wehave _ — =— ; 

or, by simplifying, cos a . tan #=sin a . E. ; 

which is a homogeneous equation, each of its terms being 
composed of two linear factors. 

The same adaptation of the formula may be more readily 
effected by merely introducing the proper power of R, as a 
factor, where it is necessary, to make the same number of 
linear factors in each term — when the formula is cleared of 
fractions, or radicals, if it contains any. 



BOOK II. 



23 



Thus in the formula 

cos 2 <z+sin 2 a=l, 

the first two terms contain two linear factors each, cos a . 
cos a and sin a . sin a, while the third term, 1, contains none. 
"We therefore multiply the 1 by R 2 or R . R ; 

cos 2 0+sin 2 a=R 2 ; 

which makes the equation homogeneous. 



(2§.) Sine and Cosine of the Sum and Difference of Two 

Arcs. 

From the sines and cosines of two given arcs may be 
obtained the sines and cosines of the sum and difference of 
those arcs ; and from these most of the necessary Formulas 
of Trigonometry may be derived. 



Let the arc AB be represented by a, 
and the arc BD by b; then AD, 
represented by a+b, is the sum, and 
a—b represents the difference, of the 
two arcs. 

Draw the sines BF, DI, and DO ; also 
draw the perpendiculars IG and IK. 




c^o 



G-F 



The triangles CBF and CIG are similar (G. 218), and give 

: BF, sin«, : IG; 



CB, R, 
CB, R, 



CI, cos b, 
CI, cos b, 



CF, cos a, 



CG. 



The triangles CBF and Dili are similar (G. 224) and give 



CB, R, 
CB, R, 



DI, sin b, 
DI, sin l, 



CF, cos a, 
BF, sin a, 



DK; 
IK. 



From these four proportions, regarding R as one or unity, 
we find • 

IG=sin a . cos b ; CG=cos a . cos b ; 
DK=cos a . sin b ; IK=sin a . sin b. (Geom. 176.) 

2* 



24 PLANE TRIGONOMETRY. 

Now IGr+DK is equal to DO, the sine of the arc AD ; and 
CG— IK is equal to CO, the cosine of AD (16... 5) ; for the sum 
of the two arcs a and b we have therefore, 

1. sin (#+5)=sin a . cos 5+cos a . sin b ; 

2. cos (a+b)= cos a . cos b— sin a . sin b. 

By changing 5 to— b, the sin 5 becomes— sin &, while cos 5 
remains the same (21) ; hence for the difference of the two 
arcs a and b we shall have 

3. sin (a— b)=sin a . cos 5— cos a . sin h ; 

4. cos (<z— Z>)=cos & . cos 5-f-sin a . sin b (Alg. 42). 

In the preceding figure each of the two arcs AB and BD, 
even the sum of these arcs, is less than a quadrant. A similar 
demonstration, however, will show that the formulas obtained 
are true, whatever be the magnitudes of the two arcs. 



(29.) Sine and Cosine of a Double Are. 

If in the preceding formulas for the sine and cosine of 
(a+b) we suppose the two arcs a and b to be equal to each 
other, we shall find 

sin 2 a=2 sin a . cos a ; 
cos 2 &=cos 2 a— sin 2 a. 



(30.) Sine and Cosine of Half a Given Arc. 

In the last two formulas suppose the arc a to be reduced 
to \ a ; the two formulas then become 

sin a=2 sin \ a . cos \ a ; 
cos a=co& 2 J a— sin 2 \ a. 

Square of radius l=cos 2 \ &+sin 2 ^ a (26.. .6) ; 
then, 1+cos a=% cos 2 J a ; 
1— cos a=2 sin 2 ^ a. 

From the last two equations we shall find 



PLANE TRIGONOMETRY. 25 



cos I a=^J i+i cos a ; 
sin i a=^J\— i cos a. 

(31.) Products of Sines and Cosines. 

For the sine and cosine of the sum and difference of two 
given arcs, a and 5, we have already found (28), 

1. sin (<z-f5)=sin a . cos 5+cos a . sin 5 ; 

2. cos (<z+5)=cos # . cos b— sin & . sin b ; 

3. sin (a— 5)=sin # . cos b— cos a . sin b ; 

4. cos (&— 5)=cos & . cos 6+sin a . sin J. 

By Addition and Subtraction among these equations we 
find 

5. sin (a+b)+sm (a— b) =2 sin «.cos5; 

6. sin (&+£)— sin (a— b)—2 cos # . sin 5; 

7. cos (&+&)+ cos (a—b) =2 cos & . cos 5 ; 

8. cos (a— b)— cos (a-f 5)=2 sin # . sin 5; 

Let (a+b) be equal to c, and (a—b) be equal to d; then 
#=2 (tf+d) ; ?=i (c— <#) ; 

and by the substitution of values in the last four formulas, 
we obtain 

9. sin c+sin d=2 sin | (c+d?) cos J (c— <#) 

10. sin c— sin d=2 cos ^ (<?+<#) sin i (c—d) 

11. cos c+cos <#=2 cos ^ (c+e£) cos \ (c—d) 

12. cos ^— cos c=2 sin ^ (<?+^) sin \ (c—d). 

If in these last formulas we make d equal to 0, then,— sin d 
being 0, and cos d being equal to the radius one (22.. .1), — 
we shall have 

13. sin c=2 sin \ c . cos I c ; 

14. cos c+l=2 cos 2 -J (?; and 1— cos c=2 sin 2 J <?. 



What follows in Articles (32), (33), and (34), 
belongs to the general theory of trigonometrical lines, but 
has no application in any subsequent portion of the present 
treatise. 



26 BOOK II. 

(32.) Various delations of the Sine and Cosine of two 
different Arcs. 

1. Divide each member of equation 1st, Article (31), by 

cos a . cos b— substitute tang, for sine— cosine (26. ..1), and the 

result will be 

sin (a+b) 

- r=tans; a+tang o, 

cos a , cos b ° & 

2. Divide each member of equation 3d, Article (31), by 
cos a , cos b— substitute tang, for sin-i-eosin (26.. .1), and there 

will result 

sin (a—b) 9 
^— tang a— tans; b. 

cos a . cos 6 ° ° 

3. Divide the 1st equation of Article (31) by the 3d — 
divide the numerator and denominator of the second member 
of the resulting equation by cos a . cos b — substitute tang, for 
sin— cos, (26... 1), and there will result 

sin (a+b) tang a+tang b 
sin {a—b) ~tang a— tang b 

4. Divide the 9th equation of Article (31) by the 10th — 
substitute tang, for sine— cos, and the reci/procal of the tang. 
for cos— sine, and there results 

sin c+sin d tang J (c+d) 
sin c— sin d ~~ tang i (c—d) 

5. Divide the 11th equation of Article (31) by the 12th — 
consider that, for radius one, the cos^-sine is equal to the cot 
(26.. .3) and also equal to the reciprocal of the tangent, and 
there will result 

cos c+cos d cot I (c+d) 
cos<$— cos c ~~ tang \ (c—d) 

6. Substitute c+d for a in the formula sin «=2 sin J a . cos 
\ a (30) — divide formula 9th of Article (31) by the resulting 
equation, and there will result 






PLANE TRIGONOMETRY. 27 

sin c+sin d cos i (c—d) 
sin (c+d) ~ cos i (c+d) 

7. Substitute c-j-d for a in the formula sin a=2 sin \ a . cos 
i a, as before — divide formula 10th of Article (31) by the 
resulting equation, and there will result 

sin c— sin d sin i {c—d) 
sin {c+d) ~sin i (c+d)" 

(33.) Tangent and Cotangent of the Sum and Difference 
of two Arcs. 

1. By substituting a+b for a in the equation cos a . tang a 
=sin a (26.. .1), we shall find 

sin (a+&) 

tang (a+b)— -, — r-rf- 

& v y cos (a +6) 

Substitute the values of this sine and cosine from the 1st 
and 2d equations of Article (31), and the second member of 
the present equation will be 

sin a . cos 5+sin b . cos a 
cos a . cos b— sin a . sin b' 

Substitute cos . tang for sine (26. ..1) — divide the resulting 
numerator and denominator by cos a . cos 6, and it will be 
found that 

- _ tang a+tang b 
tang (a+ ft)= I - i -_ 1 _ ? . 

By substituting $— J for a in the equation cos # . tang a= 
sin &, and taking the values of sin (a—b) and cos (a—b) from 
the 3d and 4th equations of Article (31), it will, in like 
manner, be found that 

7 N tang a— tang b 

tang (a— J)=— f --£_, 

& v y 1+tana.tan J 

2. By substituting #+&, and a— J, for a in the equation 
sin a . cot a=cos a (26. ..3), and proceeding in a manner 
similar to that for the tang, it will be found that 



28 BOOK II. 

, , ,. cot a . cot 5—1 

cot (a+b)= — j — — T-- 

v ' cot a+cot b 

, , 7 . cot a . cot 5+1 

and cot (a—o)= —i — • 

v J cot b— cot a 



(34.) Tangents and Cotangents of Multiple Ares. 

If in the preceding value of tang, (a+b) we make o equal 

to a, we have 

2 tanff a 

tang 2a=z — — ~ — 

& 1— tan 2 a 

Making b equal to 2a, we have, from the same value of 

tang, (a +b) 

tans: &-htang 2a 

tang Ba= T -f . b . 

° 1— tan#.tan2a 

By substituting the value of tang 2# from the preceding 

equation, and reducing the result to its simplest form, it will 

be found that 

3 tan a— tan 3 & 
tang3«= 1 _ 3tan2 -- 

By making b equal to a in the preceding value of cot (a-\- 
b), and then b equal to 2a, we may readily obtain expressions 
for the values of cotang 2a, and cotang 3a. 



PLANE TRIGONOMETRY. 29 



NATURAL AND LOGARITHMIC SINES, TANGENTS, &c. 

I. Natural Sines, Tangents, &c. 

(35.) The natural Sine of an arc is the numerical value of 
its sine when measured by the radius of the circle as unity. 
In like manner are denned the natural tangent, cosine, &c. 

To compute natural Sines and Cosines. It is evident that 
an arc may be taken so small as to differ insensibly from its 
sine ; thus an arc of one minute may be shown to be equal 
to its sine to ten places of decimals. 

The length of 1 minute on the circumference of a circle 
whose radius is 1, that is, the natural sine of 1 minute, will 
be found by dividing the circumference, 2 m (Geom. 264), by 
360X60; thus 

The natural sine of 1 minute is 

6.2831852~-21600=.0002908882. 

The natural cosine of 1 minute is 

V'l-sin 2 1 1 =.9999999577 (26...6). 

Having found the sine and cosine of 1 minute, we may 
compute the sines and cosines of 2', 3', 4', and so on, by 
means of the formulas 

sin (a+b) =2 sin a . cos 5— sin (a— b); 

cos (a+b)~ 2 cos a . cos I— cos (a—b), (31, ..5 and 7). 

Let a be equal, successively, to 1', 2', 3', 4', &c, and I be 
constantly equal to 1 ; ; the formula for sin (a+b) will give, 
successively, 



30 BOOK II. 

sin 2'=2 sin 1' cos 1'— 'sin 0=.000582 ; 
sin 3'=2 sin 2' cos l'-sin 1'=.000873 ; 
sin 4'=2 sin 3' cos l'-sin 2'=.001164; &c. 

The forruula for cos {a -\-b) will give, successively, 

cos 2'=2 cos V cos l'-cos 0=.999999 ; 
cos 3'=2 cos 2' cos l'-cos ±'=.999999 ; 
cos £'=±2 cos 3' cos l'-cos 2'=.999999 ; &c. 

In this manner the natural sines and cosines of arcs to 90° 
might be computed ; but the sine of any arc greater than 45° 
would be known from its being the same as the cosine of an 
arc as much less than 45°, — and vice versa. 

Thus the sine of 50° is equal to the cosine of 40°, and the 
cosine of 50° is equal to the sine of 40°, (16.. .4). 

The sine and cosines, or any other trigonometrical lines, of 
arcs above 90°, will be known from those of arcs to 90°. 

Tims the sine of 100°=sin (180°-100°)=sin 80° (20). 
The sine of 200°=sin (180°-200 o )=sin~20°=-sin 20°, 

(21). 



(36.) To compute natural Tangents and Cotangents, natural 
Secants and Cosecants. 

These may all be obtained from the sine and cosine ; thus 

tang =sin-r- cos ; cot=cos-7-sin ; 
sec=l-f-cos; cosec=l-f-sin (26). 

Table of Natural Sines and Cosines, 

(a 1 ?.) Table II., appended to this work, contains the natural 
sines and cosines of arcs expressed in degrees and minutes. 
The natural tangents, cotangents, &c, as already shown (36), 
may be readily obtained from the sines and cosines. 

When the number of degrees is less than 45, — find the 
degrees at the top of one of the pages, and the minutes in the 
left hand column; then opposite to the minutes, in the 



PLANE TRIGONOMETRY. 31 

column marked N. sine, or i\T. cos, at the top, will be found 
the natural sine or cosine. When the number of degrees is 
45 or more, — find the degrees at the bottom of the page, the 
minutes in the right-hand column, and then take the natural 
sine, or cosine, from the column marked JOF. sine, or iV". cos,, 
at the bottom. 

It must be understood that the numbers in the columns 
N. sine, and E". cos. are decimals — being less than the 
radius 1. 

The natural sine of 0° 30', found as above directed, is 
.00873 ; and the natural sine of 89° 20', on the same page, is 
.99993. 

(38.) To find the Natural Sine or Cosine for Degrees, 
Minutes, and Seconds. 

Find for the degrees and minutes, from the Table, as above 
directed ; take the difference between the sine, or cosine, thus 
obtained, and that which corresponds to 1' more. 

60 : this difference ; ; the given number of seconds : the 
proportional -part, to be added to the number first taken from 
the table for a sine, but subtracted from it for a cosine. Thus 

To find the natural cosine of 89° 20' 40". 

Cosine of 89° 20', is .01164, 

« of 89° 21', is .01134; diff. .00030. 

Then 60 ; .00030 \ \ 40 : .00020, proportional part for 40". 

The sines increase with the arc, from 0° to 90°, but the 
cosines diminish ; the part found for the 40" must therefore 
be subtracted ; hence 

Nat. cosine of 89° 21' 40" is .01164-.00020=.01144. 



II. Logarithmic Sines, Tangents, &c. 

(39.) The logarithmic sine of an arc is the logarithm (1) 
of the numerical value of its sine when the radius of the circle 
is regarded as 10 10 , that is, 10 000 000 000. In like manner 
are defined the logarithmic tangent, cosine, &c. 



32 BOOK II. 

The radius 10 10 is assumed when the logarithms of the 
trigonometrical lines are to be used, in order that the charac- 
teristics of these logarithms may all be positive. 

The natural sines and cosines, and the natural tangents 
below 45°, are expressed in decimals, being less than the 
radius, which is their measuring unit (35) ; their logarithms 
would therefore have negative characteristics (4). These 
negative characteristics are avoided by increasing the 
numerical expressions for all the lines in the ratio of 10 10 . 

Thus the natural sine of 1', as heretofore found, is .0002908882'; 
the logarithm of which is 4.463726. 

To avoid the— 4 we add to it 10, which is the logarithm of 
10 10 , and thus obtain the logarithm 6.463726. This last 
number is therefore the logarithm of (the natural sine of 
1'XlO 10 ), (8...1), and is called the logarithmic sine of V. It 
is the logarithm of the sine of 1', when the radius of the 
circle is 

10io=10 000 000 000. 

In this manner may be found the logarithmic sines, 
cosines, &c, of arcs, when the natural sines, cosines, &c, 
have first been computed. 

Table of Logarithmic Sines, Tangents, dec. 

(40.) Table II., appended to this work, contains the logar- 
ithmic sines, cosines, tangents, and cotangents of arcs expressed 
in degrees and minutes. The secants and cosecants, as here- 
after shown, may be readily obtained from the cosines and 
sines. 

When the number of degrees is less than 45, — find the 
degrees at the top of one of the pages, and the minutes in 
the left-hand column ; then opposite to the minutes, in the 
column marked L. sine, or L. cos, &c, at the top, will be 
found the logarithmic sine, or cosine, &c. When the number 
of degrees is 45 or more, — find the degrees at the bottom of 
the page, the minutes in the right-hand column, and then 
take the log. sine, or cosine, &c, from the column marked 
L. sine, L. cos., &c, at the bottom. 



PLANE TRIGONOMETRY. 33 

The characteristics of these logarithms are only given at 
intervals in the Table ; they must be prefixed to the mantissas 
which stand below them. 

Thns the logarithmic sine of 0° 30' is 7.940842 ; 
the logarithmic sine of 89° 20' is 9.999971. 

The numbers in column D, after the sines, in this Table, are 
the increments of the log. sines corresponding to an incre- 
ment of 1" in the arcs ; those in column D, after the cosines, 
are the decrements of the log. cosines for an increment of 1" ; 
those between the tangents and cotangents are the incre- 
ments of the tangents, and decrements of the cotangents. 
, These numbers are obtained by dividing the increments, 
or decrements, for 1 minute, by 60, the number of seconds in 
1'. The figures on the left of the decimal point, in these 
numbers, are to be understood as decimals, with 0s prefixed 
to make six of such decimal figures. 



(41.) To find the Logarithmic Sine, or Tangent, dec, for 
Degrees, Minutes, and Seconds. 

1. Find for the degrees and minutes, from the Table, as 
above directed ; multiply the corresponding number in the 
next column D, by the given number of seconds, and point off 
decimals. 

2. For a sine or tangent, add the figures on the left of the 
decimal point in the product to the logarithm already found 
(as if both were integers) — increasing the right-hand figure 
added by a unit when the adjacent rejected one is .5 or more. 
For a cosine or cotangent, subtract the product which should 
be added for a sine or tangent. 

To find the logarithmic sine of 35° 20' 45". 

The log. sine of 35° 20', as found in the Table, is 9.762177. 
The corresponding number in column D is 2.97, which, 
multiplied by 45, gives the product 133.65. We have then 
134 to add to the logarithm already found. 

Hence the log. sine of 35° 20' 45" is 9.762311. 



34 BOOK II. 

(42.) To find the Degrees, Minutes, and Seconds for any 
given Logarithmic Sine or Tangent, <&c. 

1. Find the given logarithm, if possible, in the proper 
column in the Table, observing that the columns are named 
at both the top and the bottom of the table. When the name 
of the logarithm is at the top, the degrees will be at the top, 
and the minutes opposite to the logarithm in the left-hand 
column ; when the name is at the bottom, the degrees will be 
at the bottom, and the minutes in the right-hand column of 
the Table. 

2. "When the given sine or tangent, &c, is not contained, 
in the Table, — find the degrees and minutes for the next less 
sine, or tangent, &c, in the Table ; subtract this logarithm 
from the given one ; divide the remainder (regarded as an 
integer) by the corresponding number in the next column D. 
The quotient will be the number of seconds, to be added for 
a sine, or tangent, but subtracted for a cosine, or cotangent. 



To find the degrees, minutes, and seconds corresponding 
to the logarithmic cosine 9.619300. 

The next less logarithmic cosine found in the Table — the 
name of the column being taken at the bottom of the page — 
is 9.619110, corresponding to 65° 25'. 

9.619300 
9.619110 



4.60) 190.00 (41" 

We subtract the cosine found in the Table from the given 
one, annex 00 to the remainder, and divide by 4.60, the 
number in column D corresponding to the tabular cosine. 
We thus obtain 41", to be subtracted from the 65° 25". The 
required arc or angle is therefore (65° 25')-41"=65° 24' 19". 

The 41" must be subtracted, since the arc diminishes as 
the cosine increases. 



PLANE TRIGONOMETRY. 35 

(43.) To find a logarithmic secant, subtract the logarithmic 
cosine fronu 20 ; and to find a logarithmic cosecant, subtract 
the logarithmic sine from 20. 

This results from the equations 

sec=K 2 -f-cos; cosec=K 2 -i-sin (24.. .2 and 4). 

The logarithm of the radius 10 10 is 10 ; the log. of E 2 is 
therefore 20 (8. ..3) ; hence the methods of finding a loga- 
rithmic secant and cosecant. Conversely, 

To find the degrees, minutes, and seconds, corresponding to 
a given logarithmic secant or cosecant, when the Table does 
not contain columns of these logarithms. — Subtract the secant 
from 20, to find the logarithmic cosine, or subtract the 
cosecant from 20, to find the logarithmic sine / then find the 
degrees, &c, corresponding to this cosine, or sine (42). 



RELATIONS OF THE SIDES OP PLANE TRIANGLES TO THE SINES, 
TANGENTS, &c, OF THE ANGLES. 

The sine, or tangent, &c, of an angle is the sine, or tangent, 
&c, of the arc which measures the angle. This measuring 
arc may be described with any radius / the length of the 
trigonometrical line being greater or less, in proportion to the 
length of the radius. 

THEOREM I. 

(44.) In every right-angled triangle, Radius is to the 
hypothenuse, as the sine of either of the acute angles is to the 
opposite perpendicular, or as the cosine is to the adjacent 
perpendicular. 

Let the triangle 'ABC be right- 
angled at B. 

From A, as a centre, with any radius, 
as AD, describe the arc DF between 
the sides of the angle A ; and draw DG 
perpendicular to AB. Then DG is the 
sine, and AG is the cosine, of the 
angle A. 




36 BOOK II. 

From the similar triangles we have 

AD : AC : I DG : BC, or E : AC : : sin A : BC; 

ad : ac : : AG : ab, or e : ac : : cos A : ab. 

Therefore, in any right-angled triangle, &c. 

(45.) Cor. Either of the two perpendicular sides of a right- 
angled triangle is equal to the hypothenuse multiplied by 
the natural sine of the angle opposite, or the natural cosine 
of the one adjacent, to that perpendicular. 

From the first of the preceding proportions we have 

E . BC= AC . sin A, or AC . cos C (15 and 16...4). 
When the radius E is made unity, this eq^wtion becomes 
BC=AC . nat. sin A, or AC . nat. cos (15). 



THEOREM II. 

(46.) In every right-angled triangle, ^Radius is to either 
perpendicular as the tangent of the adjacent acute angle 
is to the other perpendicular, or the secant of the sarm $ngle 
to the hypothenuse. 

Let the triangle ABC be right- 
angled at B. 

From A, as a centre, with any ra- 
dius AD, describe the arc DF between 
the sides of the angle A ; and draw DG 
perpendicular to AB. Then DG is the 
tangent of the arc DF, or of the angle 
A, and AG is the secant. 

From the similar triangles we have 

AD : AB ; : DG : BC, or E : AB ; : tan A : BC ; 

ad : ab : : AG : AC, or e : ab : : sec a : AC. 

Therefore, in any right-angled triangle, &c. 

(47.) Cor. Either of the two perpendicular sides of a right- 




PLANE TRIGONOMETRY. 37 

angled triangle is equal to the natural tangent of the opposite 
acute angle multiplied by the other perpendicular ; and 

The hypothenuse is equal to either perpendicular multi- 
plied by the natural secant of the acute angle adjacent to 
that perpendicular. 

From the preceding proportions we have 

K . BC= AB . tang A ; and K . AC= AB . sec A. 

When the radius E is made unity, these equations become 
BC=AB . nat. tan A ; and AC=AB . nat. sec A. 

THEOREM III. 

(4§.) In every plane triangle, any two of the sides are 
proportional to the sines of the angles which are respectively 
opposite to those sides. 

In the triangle ABC let DC be 
drawn perpendicular to AB, pro- 
duced when necessary. 

Then in the two right-angled tri- 
angles ADC and BDC, 

K : AC : : sin A : DC, and K : BC : : sin B : DC (44). 
From these two proportions we obtain 

AC . sin A=E . DC=BC . sin B (G. 176), 
which gives AC : sin B ; : BC : sin A (G. 180). 
Therefore, in every plane triangle, &c. 

THEOREM IV. 

(49.) In every plane triangle, the sum of any two of the 
sides is to their difference, as the tangent of half the sum of 
the angles opposite to those sides is to the tangent of half their 
difference. 




38 



BOOK II. 




In the triangle ABC let the side 
AC be less than CB. On AC pro- 
duced take CD equal to CB, and 
join BD. On CB take CF equal 
to AC ; draw AF, and produce it 
toK 

AD is the sum, and FB is the 
difference, of the two sides AC and 
CB. The angles CAF and AFC 
are equal (Geom. 48) ; and since 

the sum of these two angles is equal to the sum of CAB and 
ABC — each of these two sums being equal to the supplement 
of ACB (15)— the angle CAF or CAK is half the sum of 
CAB and ABC ; also BAK is half the difference of the angles 
CAB and ABC, because when added to their half sum it 
makes up the greater angle CAB (Alg. 168). 

The angle ADK is equal to FBK, since the lines CD and 
CB are equal ; BFK is equal to AFC or DAK, and hence 
AKD and FKB are equal (Geom. 42) ; so that the triangles 
AKD and FKB being similar, we have 



ad : fb : : kd : KB. 

But KD is the tangent of CAK, and KB is the tangent of 
BAK, for the same radius AK; therefore, in every plane 
triangle the sum of any two sides, &c. 



THEOEEM V. 



(50.) In every plane triangle, the square of any side is less 
than the sum of the squares of the other two sides by twice 
the product of those two sides into the cosine of their included 
angle, when the radius is unity. 



In the triangle ABC, in which 
the angle B is acute, we have 



AC 2 -=AB 2 -fBC 2 -2AB 

(Geom. 151). 



BD 




±» L A N E TRIGONOMETRY. 39 

But the right-angled triangle BDC gives BD equal to BO . 
cos B, when the radius is unity (45) ; hence 

AC 2 =AB 2 +BC 2 -2AB . BC . cos B. 

When the angle ABC is obtuse, we have 

AC 2 =AB 2 +BC 2 +2AB . BD (Geom. 152). 

In this case cos DBO is equal to C 

— cos ABO, since the two angles at ^s 

B are the supplements of each other y^ / 

(20), and BD therefore becomes 'y^-- / 

equal to— BC . cos ABC. Hence, j^ / 

as before, A J3 i> 

AC 2 =AB 2 +BC 2 -2AB . BC . cos ABC, 

Therefore, in every plane triangle, &c. 

THEOREM VI. 

(51.) If a, b, and c represent the sides which are respec- 
tively opposite to the angles A, B, and C of any triangle, and 
S represent half the sum of the sides, radius being unity / 



J 



,(S-*)<S-,) 



be 
From the preceding Theorem 
0«= J?+c 2 -25c . cos A ; 

b 2 +c 2 -a 2 



which sfives cos A= 



2bc 




It has heretofore been shown that 2 sin 2 J A is equal to 1— 
cos A (30) ; hence 

r, • , , a „ b 2 +c 2 -a 2 Zbc+a 2 -b 2 -c 2 
2 sin 2 \ A=l = — 

2 Ibc 2bc 

The numerator of the latter fraction is equal to a 2 — (b— c) 2 ; 
and this being the difference of the squares of the quantities 

3 



40 BOOK II. 

a and (b—c), is equal to the product of their sum and 
difference ; that is, equal to 

(a+b-c) (a-b+c), (Geom. 147). 

But since a+b+c is equal to 2S, the first of these two factors 
is equal to 28— 2c, and the second is equal to 2S— 2b. 

By substituting these values in the last equation — dividing 
both sides by 2 — reducing the second member — and extracting 
the square root of each side, 



'(8-5) (S-c) 
sin ' 



iin J A=y - 



bo 
Therefore, if a, b, and c represent the sides, &c. 

(52.) Cor. In like manner we should find 



. ■ / (S-0) (S-c) 
sin | B=\/ -- J 

* V an 



'(S-«) (8-b) 
ab 



i c=y- 



SOLUTIONS OF PLANE TRIANGLES. 

(53.) The solution of a plane triangle consists in computing 
any three of its six parts, except the three sides, when the 
other three parts are given (11). 

A plane triangle is determined by two sides and the 
included angle, by two angles and a side, or by the three 
sides. The three angles do not determine the sides, since, 
with the same angles, the sides may be increased, or 
diminished, proportionably, to any extent. 

I. Bight-angled Triangles. 

(54.) In a right-angled triangle the right angle, 90°, is 
always given ; if, therefore, any other two parts, except the 



PLANE TRIGONOMETRY. 41 

two acute angles, be given, the remaining parts may be 
computed. — When one of the acute angles is given, or has 
been computed, the other will be found by subtracting the 
first from 90° (15). 

GENERAL RULE. 

For the Solution of Right-angled Triangles. 

Form a proportion between the Eaclius, the two given parts, 
and the required part, according to Theorem I. (44:) or II. 
Make the required part the last term of the proportion, and 
compute it as a fourth -proportional. 

EXAMPLE i. 

The hyjpotJienuse and an acute angle given. 

In the right-angled triangle ABC the hypothenuse AC is 
645, and the angle A is 39° 10'. It is required to find the 
angle C, and the sides AB and BC. 

The two acute angles A and C are 
together equal to a right angle ; hence 
the angle C is 

90-(39° 10')=50° 50'. 




To find the side BC. By Theorem 
I. (44), 

Eadius : AC : \ sin A : BC. 

The natural sine of the angle A, 39° 10', is .63158 (37), 
corresponding to radius one ; then, 

1 : 645 ; : .63158 : 407.36 '=BC. 

To find the side AB. By Theorem I. (44), 

E : AC : : sin c : ab ; or e : AC : : cos A : ab. 

By taking the latter proportion, and using the natural 
cosine of the angle A, which is .77531, we have 

i : 645 : : .77531 : 500.07 '=ab. 



42 BOOK II. 

Logarithmically . Trigonometrical solutions are, in general, 
most readily effected by means of logarithms ; the logar- 
ithms of the sides of the triangles being found in Table I. 
(5), and the logarithms of the sines, tangents, &c, of the 
angles in Table II. (40). Thus, 

To find the side BC. 

Eadius 10 10 log. 10.000000 

is to AC 645 2.809560 

as sin A 39° 10' 9.800427 

is to BC 407.36' 2.609987 

The second and third logarithms are added together, and 
the first is subtracted from their sum. This finds the 
logarithm, 2.609987, of BC, the required fourth proportional 
(8...1 and 2). The natural number corresponding to this 
logarithm is found to be 407.36' (7), which is therefore the 
length of BC. 

To find the side AB. 

Eadius 10 10 log. 10.000000 

is to AC 645 2.809560 

as cos A 39° 10' 9.889477 

is to AB 500.07' 2.699037. 

The operation in this case is similar to that in the preceding, 
the logarithmic cosine of the angle A being used instead of 
the logarithmic sine. 

1. In a right-angled triangle ABC the hypothenuse AC is 

235, and the angle A is 43° 25'. Find the angle C, and the 

sides AB and BC. 

Ans. C 46° 35'; AB 170.69; BC 161.51^ 

2. In a right-angled triangle ABC the hypothenuse AC is 

94.6, and the angle C is 56° 30'. Find the angle A, and the 

sides AB and BC. 

Ans. A 33° 30'; AB 78.88 ; BC 52.21. 

3. In a right-angled triangle BDF the hypothenuse BF is 




PLANE TRIGONOMETRY. 43 

127.9, and the angle B is 40° 10' 30". Find the angle F, and 
the sides BD and DF. 

Ans. F 49° 49' 30" ; BD 97.72 ; DF 82.51. 

EXAMPLE n. 

The hypothenuse and a perpendicular side given. 

In the right-angled triangle ABC the hypothenuse AC is 
480, and the side AB is 288. It is re- 
quired to find the two acute angles A 
and C, and the perpendicular BC. 

To find the angle A. By Theorem I. 
(44), we have 

Kadius : AC : : cos A : AB. 

By inversion of the terms, to make the required angle A 
the last term, we have 

AC : Radius : : AB : cos A (Geom. 182). 

AC 480 comp. log. 7.318759 

is to radius 10 10 10.000000 

as AB 288 2.459392 

to cos A 53° T 49" 9.778151. 

The logarithm of 480 is found in the Table to be 2.681241 ; 
the complement of this logarithm is 7.318759, which is added 
up with the second and third logarithms, 10 being rejected 
from the sum (9). 

The logarithmic cosine thus obtained is to be sought for in 
Table II. The nearest less cosine found in the table is 
9.778119, corresponding to 53° 8'. 

To find the seconds to be subtracted, we subtract this cosine 
from the one in question, annex 00 to the remainder, and 
divide it by the number 2.81 in column D. 

2.81) 3200 (ll". 

The required value of the angle A is therefore 



44 BOOK II. 

53° 8 / -ll // -=53° 7 49", (42). 

To find the angle C. Subtract the angle A from 90° ; thus 
the angle O=90°-(53° 7' 49")=36° 52' ll". 

To find the perpendicular BC. We might employ the 
proportion. 

Eadius : AC ; ! sin A : BO. 

But BC may be found by extracting the square root of the 
difference of the squares of the sides AC and AB ; thus 

BC= 7480^288^=384 (Geom. 149). 

4. In a right-angled triangle ABC the hypothenuse AC is 
340, and the side AB is 200. Find the acute angles A and 
C, and the perpendicular BC. 

Ans. A 53° 58' 6"; C 36° 1' 54"; BC 274.95. 

5. In a right-angled triangle ABC the hypothenuse AC is 
95.75, and the side BC 60. Find the acute angles A and.C, 
and the perpendicular AB. 

Ans. A 38° 48' 7" \ O 51° 11' 53" ; AB 74.62. 

6. In a right-angled triangle BDF the hypothenuse BF is 
3470.8, and the side BD 2000. Find the acute angles B and 
F, and the perpendicular DF. 

Ans. B 54° 48' 50"; F 35° 11' 10"; DF 2836.63. 

EXAMPLE III. 

A perpendicular side and an acute angle given. 

In the right-angled triangle ABC the side AB is 200, and 
the angle A is 34° 45 '. It is re- 
quired to find the perpendicular 
BC, and the hypothenuse AC. 

To find BC. By Theorem II. 
(46), we have 

Eadius : AB : : tang A ; ; BC. 




PLANE TRIGONOMETRY. 45 

Eadius 10 10 10.000000 

is to AB 200 2.301030 

as tang A 34° 45' 9.841187 

is to BC 138.74 2.142217. 



To find AC. By Theorem II. (46), we have 
Eadius : AB ; ; secant A : AC. 

The logarithmic secant of the angle A — not being given in 
Table II. — would be obtained by subtracting the logarithmic 
cosine of A from 20 (43). 

But, without using a secant, AC may be found from the 
proportion 

cos A : AB : : Eadius : AC (44). 

Or, AC is equal to the square root of the sum of the squares 
of AB and BC ; thus 

AC= V200 2 +138.74 2 =243.41 (Geom. 148). 

7. In a right-angled triangle ABC, the side AB is 249, and 
the angle A is 29° 14', Find the perpendicular BC, and the 
hypothenuse AC. 

Ans. BC 139.35 ; AC 285.341. 

8. In a right-angled triangle ABC the side BC is 364.3, 
and the angle A is 50° 45'. Find the perpendicular AB, and 
the hypothenuse AC. 

Ans. AB 297.645 ; AC 470.433. 

9. In a right-angled triangle BDF the side BD is 39.235, 
and the angle F 60° 15'. Find the perpendicular DF, and 
the hypothenuse BF. 

Ans. DF 22.424 ; BF 45.191. 




46 BOOK II 



EXAMPLE IV. 

The two perpendicular sides given. 

In the right-angled triangle ABO 
the perpendicular AB is 239, and BC 
188. It is required to find the acute 
angles A and 0, and the hypothenuse 
AC. 

To find the angle A. By Theorem 
II. (46.) 

Eadius : AB : ; tang A : BC. 

By inversion of the terms, to make the required angle A 
the last term, 

AB : Kadius : : BO : tang. A (Geom. 182). 

AB239 comp. log. 7.621602 

is to radius 10 10 10.000000 

as BC 188 2. 274158 

is to tang A 38° 11' 20" .... 9.895760. 

We take the arithmetical complement of the logarithm of 
239 ; the second and third logarithms are added to this 
complement, and 10 is rejected from the sum. Thus is 
obtained the logarithmic tangent of A, to which correspond 
38° 11' 20" (42). 

The angle C=90°-(38° 11' 20")=51° 48' 40". 

The hypothenuse AC, found as in Example III., is 304.08. 

10. In a right-angled triangle ABC the perpendicular AB 
is 736.3, and BC 500. Find the acute angles A and C, and 
the hypothenuse AC. 

Ana. A 34° 10' 45" ; C 55° 49' 15" ; AC 890.02. 

11. In a right-angled triangle ABC the perpendicular AB 



PLANE TRIGONOMETRY. 



47 



is 300, and BO 273.84. Find the acute angles A and C, and 
the hypothenuse AC. 

Ana. A 42° 23' 23" ; C 47° 36' 37" ; AC 406.18. 

12. In a right-angled triangle BDF the perpendicular BD 
is 246.32, and DF 380.07. Find the acute angles B and F, 
and the hypothenuse BF. 

Ans. B 57° 3' 11" ; F 32° 56' 49" ; BF 452.91. 

II. Solutions of any Plane Triangles. 

The methods of solution under the preceding Examples are 
applicable only to right-angled triangles, and for such tri- 
angles those methods are, in general, to be preferred to any 
others. The following are applicable to every kind of plane 
triangles, and provide for every case in which a triangle is 
determined by three given parts (53). 

CASE I. 

(55.) When a Side and the opposite Angle are two of the 
given parts. 

The solution of a triangle in this case depends on the 
principle that the sides of any plane triangle are proportional 
to the sines of the opposite angles (48). 



EXAMPLE. 



In the triangle ABC, the angle A is 46° 30', the side AB 
is 240, and BC 200. It is required to find the angles B and 



C, and the side AC. 



It is evident that there may be 
two different triangles having the 
given angle A, and the given sides 
AB and BC. In on£ of these two 
triangles the required angle ACB 
is acute, in the other it is obtuse; 
and it will accordingly be found 
3* 





48 BOOK II. 

that the solution finds two different 
values for this angle. 

To find the angle ACB. By The- 
orem III. (48), we have 

BC : sin A : ; AB : sin ACB. 



Logarithmically. BC 200 2.301030 

is to sin A 46° 30' ... . 9.860562 

as AB 240 2.3 80211 

is to the sine of ACB 60° 30' . . . 9.939743. 



The first logarithm is subtracted from the sum of the 
second and third, which leaves the logarithmic sine 9.939743 
of the required angle ACB. The number next less than 
this among the logarithmic sines in Table II. is 9.939697, 
corresponding to 60° 30', which, without computing the addi- 
tional seconds, we take for one of the values of the angle ACB. 

Every sine corresponds to two arcs which are supplements 
of each other (20) ; and the supplement of 60° 30 / is 

180°- (60° 30')=119° 30'. 

In the solution of this triangle we therefore find the angle 
ACB to be 

60° 30', or else 119° 30'. 

When the angle ACB is acute, the third angle ABC is 
equal to 

180°-(46° 30'+60° 30')=73 c 



>o. 



but when the angle ACB is obtuse, the third angle ABC is 
equal to 

180°-(46° 30'+119° 300=14° (Geora. 38). 

To find the side AC, when the angle ACB is acute. By 
Theorem III., we have 

sin A : BC : : sin ABC : AC. 



PLANE TRIGONOMETRY. 49 

The value of the angle ABC corresponding to the acute 
angle ACB is 73° ; then 

sin A 46° 30 / eomp. log. 0.139438 

is to BC 200 2.301030 

as sin ABC 73° 9.980596 

is to AC 263.67 2,421064 

In like manner might be found the value of the side AC 
when the angle ACB is obtuse ; the angle ABC in that case 
being 14°. 

In reference to the present case, in the Solution of Tri- 
angles, observe that 

(56.) When the given angle of a triangle is acute, and 
opposite to the less of the two given sides ; the angle opposite 
to the other given side, found from its sine, has two values 
which are supplementary to each other (20), and two different 
triangles satisfy the given conditions ; but 

When the given angle is not acute, or is not opposite to the 
less of the two given sides ; the angle opposite to the other 
given side can only be acute, and there can be but one triangle 
satisfying the given conditions. For 

If the given angle be right or obtuse, each of the required 
angles must be acute (Geom. 39) ; and if the given side, as 
BC, opposite to the given angle A, were equal to, or greater 
than, the other given side AB, it is evident that BC could 
have but one position, on the same side of AB, and that it 
would make an acute angle with AC. 

13. In a triangle ABC, the side AB is 98, the side BC 
95.12, and the angle C 33° 21'. Find the angles A and B, 
and the side AC. 

The given angle not being opposite to the less of the two 
given sides, there can be but one triangle satisfying the given 
conditions (56). 

Am. A 32° 14' 55" ; B 114° 24' 5" ; AC 162.33. 



50 BOOK II. 

14. In a triangle ABC, the angle A is 79° 21', the angle 
B 54 22', and the side BC 125. Find the angle C, and the 
sides AB and AC. 

The angle C will be fonnd by subtracting the sum of the 
two given angles from two right angles or 180°. 

Ans. C 46° 17' ; AB 91.92 \ AC 103.37. 

15. In a triangle ABC, the side AB is 254,3, the side AC 
396.8, and the angle B 94° 29'. Find the angles A and C, 
and the side BC. 

For the sine of the obtuse angle B take the sine of its 
supplement (20). 

Ans. A 45° 48' 21"; C 39° 42' 39"; BC 285.37. 

16. In a triangle BDF, the angle B is 40°, the side BD is 
400, and the side DF 350. Find the angles D and F, and 
the side BF. 

Ans. D 92° 43' 32", or 7° 16' 28" ; F 47° 16' 28", or 
132° 43 / 32" ; BF 543.88, or 68.94. 



CASE II. 
(57.) When two Sides and the included Angle are given. 

The given angle subtracted from 180° will leave the sum 
of the two required angles. Then 

The sum of the two given sides is to their difference, as the 
tangent of Ivalf the sum of the two required angles is to the 
tangent of half their difference (49). 

Half the sum of the two required angles, increased by half 
their difference, will be the greater angle, and, diminished by 
half their difference, will be the less angle (Alg. 168). 

The greater of the two angles thus found will be opposite 
to the greater of the two given sides, and the less opposite to 
the less side (Geom. 56). 



PLANE TRIGONOMETRY. 



51 




EXAMPLE. 

In the triangle ABC, the side V 
AB is 240, AC 180, and the in- 
cluded angle A is 36° 40'. It 
is required to find the angles B 
and C, and the side BC. J^ — — ^b 

The sum of the two required angles B and C is 

180°-(36° 400=143° 20' ; one half of which is 71° 40'. 

The sum of the two given sides is 420, and their difference 
is 60. 

The sum 420 comp. log. 7.376751 

is to the cliff. 60 1.778151 

as tang 71° 40' 10.479695 

is to tang, of J (C-B) 23° 19' ... . 9.634597 

The number 9.634597 is the logarithmic tangent of half 
the difference of the angles B and C. The number next less 
than this among the logarithmic tangents in Table II. is 
9.634490, corresponding to 23° 19', which, without computing 
the additional seconds, we take for half the difference of the 
angles B and C. 

Then the greater angle C is 

71° 40'+(23° 19')=94° 59'; 
and the less angle B is 

71° 40'-(23° 19')=48° 21'. 

Having now found the angles B and C, the side BC would 
be found from the proportion, 

sin B : AC : ; sin A \ BC (48). 

17. In a triangle ABC, the side AB is 103, AC 126, and 
the included angle A is 56° 30'. Find the angle B and C, 
and the side BC. 

Ans. B 72° 20' 15"; C 51° 9' 45"; BC 110.267. 



52 BOOK II. 

18. In a triangle ABC, the side AB is 304, BC 280.3, and 
the included angle B is 100°. Find the angles A and C, and 
the side AC. 

Ans. A 38° 3' 3" ; C 41° 56' 57" ; AC 447.856. 

19. In a triangle BDF, the side BF is 123.75, DF 500, and 
the included angle F 120°. Find the angles B and D, and 
the side BD. 

Ans. B 49° 12' 4" ; D 10° 47' 56" ; BD 572.006. 

case ni. 
(58.) When the three sides are given. 

Divide the triangle into two right-angled triangles by a 
perpendicular drawn from one of its vertices to the opposite 
side regarded as the base. Then 

The base is to the sum the other two sides, as the difference 
of those two sides is to the difference of the two segments 
of the base (Geom. 156 and 180). 

Half the sum of the base and the difference of its segments 
will be the greater segment ; half the difference between the 
base and the difference of its segments will be the less segment 
(Alg.168). 

The greater segment of the base will be adjacent to the 
greater of the other two sides, and the less to the less 
(Geom. 58). 

The required angles may then be found from the right- 
angled triangles. 

EXAMPLE. 

c 

In the triangle ABC, the side 

AB is 800, AC is 600, and BC 
400. It is required to find the 
angles of this triangle. 



^ B 



Draw the perpendicular CD. Then, the base AB being 
800, the sum of AC and BC 1000, and their difference 200, 
we have 



PLANE TRIGONOMETRY. 53 

800 : 1000 : : 200 : 25o=ad-db. 

Then AD is \ (800+250)=525 ; and DB is } (800-250)= 
275. 

To find the angle A. In the right-angled triangle ADC, 
we have the hypothenuse AC 600, and the perpendicular 
AD 525. 

Eadius : AC : : cos A : AD (44) ; 
or AC : radius ; ! AD ; cos A (Geom. 182). 

AC 600 .. . comp. log. 7.221849 

is to radius 10 10 10.000000 

as AD 525 2.720159 

is to cos A 28° 57' ... . 9.942008 

In like manner the angle B might be computed in the right- 
angled triangle BDC ; or the angle B may be found from 
the proportion 

BC : sin A : ; AC : sin B (48). 

The angle ACB will be found by subtracting the sum of A 
and B from 180°. 

Otherwise, 

The angles of any plane Triangle may also be found from 
the sides by the formula 



Sin i A 



v 1 



(8-b) (S-c) . 



bo 



in which A will be any required angle, S half the sum of the 
three sides, and b and c the sides containing the required 
angle (51). # 

This formula makes the radius unity / and the value of the 
second member, when computed, would express the natural 
sine of the angle A. 

To adapt this formula to the radius, E, of the logwrkhmio 
sines, we square both sides, and insert the factor R 2 in the 
second member (27) ; thus 

R2(S-&) (8-_c) 



Sin 2 1 A= 



54 BOOK II. 

Computation of the angle A in the preceding triangle : 

Log. E 2 , (8...3) 20.000000 

" S-5, 300 2.477121 

" S-c, 100 2.000000 

" b, 600 . comp. log. 7.221849 

" c, 800 . comp. log. 7.096910 

Log. sin 2 i A 18.795880 

The product be being a divisor, in the formula, we take 
the complements of the logarithms of b and c, and reject 20 
from the sum of all the logarithms (9) ; the result is the 
logarithm of the square of the sine of half the angle A. 

Then \ of 18.795880=9.397940=log. sin \ A, (8...4). The 
angle corresponding to the nearest less logarithmic sine in 
Table II. is 14° 28', which we take for half the angle A. 
Then 

A=(14° 28')X2=28° 56'. 

The value before found for the angle A is 28° 57' ; the 
difference is owing to the omission of seconds in taking the 
quantities from the table. 

By this method we find half the required angle from its 
sine. This half angle is necessarily acute, since the whole 
angle is less than 180°. "No ambiguity therefore attaches to 
an angle thus found, as is sometimes the case with an angle 
found from its sine (56). 

20. In a triangle ABC, the side AB is 460, BC is 340, and 
AC 280. Find the angles A, B, and C. 

Ans. A 47° 23' 16" ; B 37° 18' 31" ; C 95° 18' 13". 

21. In a triangle ABC, the side AB is 95.6, BC is 275, and 
AC 300. Find the angles A, B, and C. 

Ans. A 65° 47' 55" ; B 95° 42' 52" ; C 18° 29' 13". 

22. In a triangle BDF, the side BD is 500, DF is 403.7, 
and BF 395.75. Find the angles B, D, and F. 

Ans. B 52° V 3"; D 50° 34' 45" ; F 77° 25' 12". 



PLANE TRIGONOMETRY. 



55 



GRAPHIC SOLUTIONS. 

(59.) A triangle, or other polygon, may be constructed 
from given parts, determining the figure, and the required 
parts may then be measured in the figure. * This constitutes 
its graphic solution. For this method a few simple instru- 
ments are necessary. 

1. The Dividers consist of two legs, terminating 
in sharp points at one extremity, and revolving on 
a pivot which unites them at the other extremity. 

This instrument is used in laying off given 
distances on a straight line, or plane ; in describ- 
ing circles, &c. 

One of its le^s is sometimes furnished with an 
adjusting screw, by which a slow and regular mo- 
tion may be given to one of its points. It is then 
called the hair compasses. 

A part of one of its legs is sometimes remov- 
able, that other parts may be substituted, to de- 
scribe a circle with a lead pencil, or with ink, &c. 

2. The Plane Scale is a ruler containing lines of equal 
parts, chords, sines, tangents, &c. It commonly contains also 
a diagonal scale, abed, from which may be taken tenths and 
hundredths of the main divisions. 



a. 2 . 4 ■ 6 . 









i \ i 1 1 l M > i " 








M 1! . ! i U 1 








t 1 1 1 1 \ I I ! 








II 1 1 ! 1 1 1 1 I .1 








11 \\ l|MU 1 M 








II 1 l iVi i \ 1 i 








hhiiimi 








... .. huimin 






1 1 \ \ \ \ \ 1 1 1 I t 






1 hnimiil 



,03 
4 
06 
03 



rl 



To taJce with the Dividers any number of units, tenths, and 
hundredths from this Scale. — One point of the dividers at the 
proper number on the upper line of the scale and the other 
at a, will embrace the units y to obtain the tenths, move the 
latter point, to the proper division, along the line a b ; to 
obtain the hundredths, move the hitter point, from the 



56 



BOOK II 



number of tenths, down the diagonal line, to the proper 
number of hundredths, and move the other point of the 
dividers to a corresponding position in the perpendicular 
drawn from the number of units. 

3. A line of chords, often found on the plane scale, shows 
the length of the chords for every degree of the quadrant, 
proportioned to any convenient radius, — the chord of 60°, 
on this line, being always equal to the assumed radius 
(Geom. 128). 



Chords 


10 


20 m 40 50 60 70 80 90 








Sines 


10 


20 m 40 50 60 709|0 30 40 50 


jSeeants 60 


Tang 


«- 


SO 30 40 50 


60 



To lay off an Angle containing a given number of degrees, 
from a line of Chords. — "With a radius equal to the chord of 
60° describe an arc of a circle ; take with the dividers the 
chord of the given number of degrees, and point off this chord 
on that arc ; then through these two points and the centre of 
the circle draw the two sides of the angle. 

The lines of sines, tangents, and. secants show the length of 
these lines for every degree of the quadrant, proportioned to 
the same radius as the chords. The line of secants is an 
extension of that of the sines, since the secant of is equal to 
the radius, or sine of 90°. 



4. The Protractor is a 
semicircle whose arc is 
divided into degrees, and 
sometimes into half de- 
grees, for the convenient 
laying off, or measuring, 
of angles. Its centre is 
marked by a small notch 
at c. in the diameter ah. 



To lay off, with . the Protractor, an Angle containing a 
given number of degrees. — Draw a straight line for one side 




PLANE TRIGONOMETRY. 



57 



of the angle ; make the diameter ah coincide with this line, 
with the centre c at the point which is to be the vertex of the 
angle ; point off the given number of degrees from the arc 
of the protractor, and then through this point and the vertex 
at c draw the other side of the angle. 

By applying the Protractor to a given angle, with the 
centre o at its vertex, and the radius ac or cb on one of its 
sides, we may ascertain the number of degrees contained in 
the angle. 



5. The Sector consists of two equal arms, 
or sides, revolving about a pivot which unites 
them at one extremity ; with several scales 
on its faces, and diverging diagonal lines 
(represented in the accompanying figure) 
divided into equal parts. 

To take with the Dividers, from the Sector, 
a number of units, in the proportion of a 
given number to an inch. — Suppose, for ex- 
ample, that a line 7 feet long is to be repre- 
sented in the proportion of 5 feet to an inch. 
Take an inch with the dividers from the scale 
of inches ; open the sector until the two 
points of the dividers can be placed on the points 5, 5 on the 
two arms of the sector ; then open the dividers to the points 
7, 7 on the arms of the sector. 

The reason for this operation will be understood by 
regarding the lines or distances concerned as the sides of 
similar triangles. 




To lay off, from the Sector, a given number of Degrees on 
an arc of any radius. — Take the given radius in the dividers ; 
open the 'sector until the two points of the dividers can be 
placed on the points 60, 60 on the scales of chords on the two 
arms of the sector ; then adjust the points of the dividers to 
the given number of degrees on the scales of chords, and lay 
off this distance on the given arc. 

The reason for this operation will be understood from an 



58 



BOOK II. 



inspection of the scales of chords on the two arms of the 
sector — not represented in the preceding figure. 

6. The Parallel Ruler consists of two rulers connected 
together by means of bars and pivots, and having a parallel 
motion with respect to each other. 




Through a given point to draw a straight line parallel to a 
given straight line. — Make the edge of the ruler db coincide 
with the given line ; hold cd firmly to the paper, and move 
the edge of db to the given point ; hold ah firmly to the paper, 
and along its edge draw the required parallel. 

Gutter's Scale — not so much used as formerly — is com- 
monly two feet in length, containing the plane diagonal scale 
of equal parts, scales of sines, chords, and tangents, on one 
side of it ; and scales for logarithmic computation on the 
other. 



EXAMPLE. 

In the Graphic Solution of Triangles. 

In the triangle ABC, the angle A is 46° 30', the side AB 
230 feet, and the angle C 96°. Find the other parts of the 
triangle by an instrumental construction and measurement. 

The triangle must 
be constructed from 
the given parts / 
namely, the angle 
A, the side AB, and 
the angle C. 

We may take any 
convenient number 
oifeet to an inch or main division of the scale of equal parts 




PLANE TRIGONOMETRY. 59 

(59.. .2), and lay off the side AB. Taking 100 feet to an inch, 
the side AB is 

230-rl00=2.3 inches. 

Take, with the Dividers, 2.3 from the scale, and with this 
distance point off the side AB. Then, with the Protractor, 
make the angle BAG 46° 30'. 

The angle C is given, but the side AC not being given, Ave 
have not the position of the vertex C, and cannot therefore 
construct the angle C at its vertex; but 180° minus the sum 
of the angles A and C gives the angle B 37° 30', with which 
we construct the angle ABC. The meeting of the sides AC 
and BC forms the angle ACB (Geom. 277). 

The angles A and B might have been constructed from a 
Line of Chords (59.. .3). 

To measure the side AG. — Open the Dividers to the points 
A and C ; apply the dividers to the scale of equal parts, so 
as to find the number of units, tenths, and hundredths in AC ; 
multiply this number by 100, the number of feet to an inch 
in the side AB ; the product will be the number of feet in 
the side AC. The side BC will be measured in a similar 
manner. 

With an exact construction, AC will be 1.408, and BC 
1.68 on the scale ; then AC is 14:0.$ feet, and BC 168 feet. 

The required sides must always be measured with the same 
number of feet, yards, &c, to a unit of the scale as that with 
which the given. sides are laid off. 

A required angle may he measured in degrees, in a con- 
structed figure, by applying the centre of the Protractor to 
the vertex of the angle, and its diameter to one of the sides ; 
the other side of the angle will cut off the number of included 
degrees. When the sides of the angle are shorter than the 
radius of the Protractor, they will need to be produced. 

(60.) Scho. — A graphic solution cannot be relied upon with 
entire confidence, since in the construction and use of the 
instruments we cannot expect perfect accuracy. It is a con- 
venient method of obtaining an approximate result. 



60 BOOK II. 

EXEECISES 
In the several Gases of Plane Trigonometry. 

The student should be exercised in the graphic, as well as 
in the trigonometrical solutions of these problems. 

23. In the right angled triangle ABC, there are given the 
angle A 49°, and the hypothenuse AC 345.6, to find the sides 
AB and BC. Ans. AB 226.733 ; BC 260.827. 

24. In the right-angled triangle ABC, there are given the 
hypothenuse AC 700, and the side AB 259.34, to find the 
angles A and C, and the side BC. 

Ans. A 68° 15' 16" ; C 21° 44' 44" ; BC 650.186. 

25. In the right-angled triangle ABC, there are given the 
side BC 473.5, and the angle C 38° 35', to find the hypothe- 
nuse AC, and the side AB. 

Ans. AC 605.729 ; AB 377.764. 

26. In the right-angled triangle ABC, there are given the 
perpendicular AB 381.75, and BC 485.9, to find two acute 
angles, and the hypothenuse AC. 

Ans. A 51° 50' 41" ; C 38° 9' 19"; AC 617.92. 

27. In the right-angled triangle ABC, there are given the 
side AB 300, BC 400, and AC 500, to find the acute angles 
A and C (55). Ans. A 53° V 48" ; C 36° 52' 12". 

28. In the triangle ABC, there are given the angle A 50° 
45', the angle B 95° 30', and the side AB 536.75, to find the 
other two sides AC and BC. 

Ans. AC 961.68 ; BC 748.16. 

29. In the triangle ABC, there are given the side AB 400, 
the angle A 57° 15', and the side BC 350, to find the angles 
B and 0, and the side AC (56). 

Ans. B 48° 45' 55", or 16° 44' 5" ; C 73° 59' 5", or 
106° 0' 55"; AC 312.95, or 119.82. 



PLANE TRIGONOMETRY. 61 

30. In the triangle ABC, there are given the angle A 56° 
40', the side AB 240.3, and the side AC 373.65, to find the 
angles B and C, and the side BC. 

Ans. B 83° 36' 26" ; C 39° 43' 34" ; BC 314.132. 

31. In the triangle ABC, there are given the side AB 900, 
BC 436.4, and AC 600, to find the angles A, B, and C. 

Ans. A 24° 54' 24" ; B 35° 22' 54" ; C 119° 42' 42". 



TRIGONOMETRICAL COMPUTATION OF HEIGHTS 
AND DISTANCES. 

(61.) Heights and Distances to which a measuring instru- 
ment cannot be applied, may be computed as the sides of 
imagined triangles, when a sufficient number of the other 
parts of those triangles have been measured instrumentally, 
or computed from other measured triangles. 

In these applications of Trigonometry some new terms will 
be employed, which must first be defined. 

1. A vertical line is the direction of a plumb-line, that is, 
of a thread fastened at one end, and hanging freely, with a 
weight at the other end. A vertical plane is any plane 
passing through a vertical line. 

2. A horizontal line is any straight line at right angles 
with a vertical line. A horizontal plane is any jDlane at 
right angles with a vertical line or plane. 

3. A verticle angle is any angle formed in a vertical plane ; 
and a horizontal angle is any angle formed in a horizontal 
plane. 

4. An angle of elevation is a vertical angle having one side 
horizontal, and the other side above the horizontal side. An 
angle of depression is a vertical angle having one side hori- 



6: 



BOOK II. 



zontal, and the other side below the horizontal side. A 
sloping angle is one which is formed in an inclined or 
sloping plane, being neither a vertical nor a horizontal 



angle. 



Measurement of Angles. 



(62.) The angle subtended, at any assumed point of obser- 
vation, by any two objects within the reach of vision, may 
be measured with a graduated circle, placed with its centre 
at the vertex of the angle, and its plane in the plane of the 
angle. 

In the accompanying 
figure, an index line at- 
tached to the centre of a 
graduated circle, and re- 
voluble in the plane of 
that circle, is represented 
as being directed, by the 
eye, successively to the 
two distant objects B and 
D. The number of de- 
grees intercepted on the 
circumference, between 

the two lines of sight, is the measure of the angle BCD, — 
which may be either a vertical, a horizontal, or a sloping 
angle. 

The instruments which have been constructed for measuring 
angles consist of a circle, or arc of a circle, divided into 
degrees, and sometimes into parts of a degree — with various 
appendages designed to secure convenience and accuracy in 
the use of such instruments. 




Use of the Vernier. 

(63.) A vernier is a small scale adapted to measure parts of 
the divisions on another scale, or on the graduated arc of any 
circular instrument. 



PLANE TRIGONOMETRY. 63 

Its use will be understood from the following figure : 



29 



J 2 8 4 If 6 7 3 



30 

Z 2 S 4 5 



a i i u i m 



AB represents a portion of a scale divided into inches, 
and these inches into tenths of an inch — the divisions being 
here enlarged, for greater distinctness. Underneath is the 
vernier scale ed, whose 10 equal parts are together equal to 
9 tenths of an inch on AB. Each division of the vernier is 
therefore 9 hundredths of an inch, and 1 hundredth of an inch 
less than 1 tenth of an inch on AB. 

Let x he a point between 29.2 and 29.3 inches from the 
beginning of the scale AB, and let it be required to find its 
distance in inches and hundredths of an inch. 

Move the line of the vernier to the point x, and then 
observe what division line on cd coincides with a division 
line on AB. We see that 6 on cd coincides with S on AB. 
Then the point x is 6 hundredths of an inch farther along the 
scale AB than is the line 29.2 ; that is, its distance on the 
scale AB is 29.26 inches. This is evident from considering 
that the six divisions, from to 6, on cd, are 6 hundredths of 
an inch less than the six divisions, from 2 to 8, on AB. 

If the 7th line on cd coincided with a line on AB, when 
the line is moved to the point x / that point would thus be 
shown to be 7 hundredths of an inch from the line on AB 
which is next on the left of it, and so on. 



2. A vernier scale may in like manner be adapted to the 
graduated arc of any circular instrument. If the arc be 
divided to half-degrees, and 30 divisions on the vernier be 
equal to 29 half-degrees, each vernier division will be 1 minute 
less than a half-degree, and the vernier will thus measure 
minutes on the graduated arc. 




64: BOOK II, 



EXAMPLE I. 

"Wanting to know the height of a tree standing on a 
horizontal plane, I found the angle of elevation of its top, at 
the station which I then occupied, to be 26° 30'. Measuring 
75 feet directly towards the tree, the elevation of its top was 
then found to be 51° 30' ; what was the height of the tree ? 

Construction. — Draw a 
horizontal base line AD, on 
which lay off AB, 75 feet 
(59.. .2). Make the angle 
DAG 26° 30', DEC 51° 30 / 
(59. ..3 or 4), and produce 
the lines AC and BC until 
they meet in C ; the point C will represent the top of the 
tree, and CD, drawn perpendicular to AD, will represent the 
height of the tree, according to the same scale on which AB 
was measured (59. ..2). 

Computation. — Subtract the angle A, 26° 30', from the 
angle DBC, 51° 30', to find the angle ACB, 25° (Geom. 43). 

In the triangle ABC find the side BO, 79.18 feet (55) ; then 
in the right-angled triangle BDC we shall have the side BC 
and the angle DBC, to find the perpendicular DC, 61.97 
feet (54). 

The preceding solution, in actual practice, would give the 
height of the tree above the horizontal line, AD, passing- 
through the centre of the instrument with which the angles 
of elevation, DAC and DBC, are measured. To this the 
height of that centre above the plane should be added, for 
the entire height of the tree. 

EXAMPLE II. 

From the top of a tower, by the sea-side, of 143 feet in 
height, it was observed that the angle of depression of a ship's 
hull, then at anchor, measured 35°. What was the ship's 
distance from the base of the tower ? 



PLANE TRIGONOMETRY. 



65 



Construction. Draw a 
horizontal base line AC; 
draw AB, to represent the 
tower, perpendicular to 
AC, and equal to 143 feet 
(59.. .2). ; draw the horizon- 
tal line BD, parallel to AC ; 

make the angle of depression DBC 35° (59.. .3 or 4) ; — the 
intersection of the line BC with AC will represent the posi- 
tion of the ship, and AC the ship's distance from the base of 
the tower. 




Computation. The angle ACB is equal to the angle CBD, 
35° (Geom. 26) ; then in the right-angled triangle CAB, we 
have the perpendicular AB, 143 feet, and the angle C, 35°, 
to find AC, 204.22 feet. 



EXAMPLE in. 



Wanting to know the height of a steeple standing on a hill, 
at some distance from its base I found the angle subtended 
by its base and summit to be 28°. Having set up a staff 70 
feet from my station, I found the staff and base of the steeple 
to subtend 135°, and the staff and summit 105° ; then going 
to the staff, and causing that to be set up at my former 
station, the staff and the base subtended 20°, and the staff 
and summit 45°. What was the height of the steeple ? 



Let the distance be- 
tween the two stations 
be represented by the 
straight line AB, which 
may or may not be in the 
same plane with the two 
lines AC and CD. Then, 
A being the first station, 

the angle CAD is 28°, BAC 135°, BAD 105°; ABC 20°, and 
ABD 45°. 

5 




6(3 BOOK II. 

The intersection of the lines AC and BC determines the 
position of the base, and the intersection of AD and BD that 

of the summit, of the steeple. 

This problem admits of a graphic solution, by a construction 
in a plane, only when the lines BA, AC, and CD are in the 
same vertical plane ; and in that case it would not be 
necessary to measure the angle CAD, since it would be found 
by taking the difference of the angles BAC and BAD. 

In the general case of the problem, the angle BAC may be 
either a vertical, a horizontal, or a sloping angle. 

Computation. In the triangle BAD we have the side BA, 
70 feet, the angle BAD, 105°, and ABD, 45°, to find the side 
AD, 99 feet (55). 

In the triangle BAC, we shall have BA, 70 feet, the angle 
BAC, 135°, and ABC, 20°, to find the side AC, 56.65 feet; 
then 

In the triangle ACD, we shall have AD, 99 feet, AC, 56.65 
feet, and the angle CAD, 28°, to find CD, 55.73 feet (57), 
which is the required height of the steeple. 

1. Having measured 200 feet, in a direct horizontal line, 
from the bottom of a steeple, the angle of elevation of its top, 
taken at that distance, was found to be 47° 30'. What was 
the height of the steeple ? Ans. 218.26 feet. 

2. From the edge of a ditch 36 feet in width, surrounding 
a fort, having taken the angle of elevation of the top of the 
wall, it was found to be 62° 40'. What must be the length 
of a ladder to reach from my station to the top of the wall ? 

Ans. 78.4 feet. 

3. What was the perpendicular height of a balloon, when 
its angles of elevation were 35° and 64°, as taken by two 
observers, at the same time, both on the same side of it, and 
in the same vertical plane ; the distance between the two 
observers being 880 yards? Ans. 935.757 yards. 



PLANE TRIGONOMETRY. 



67 



4. Having to find the height of an obelisk standing on the 
top of a declivity, I first measured from its base a distance 
of 40 feet, and then found the angle formed by the oblique 
plane and a line imagined to go to the top of the obelisk, 
41° ; but, after measuring on in the same direction 60 feet 
farther, the like angle was only 23° 45'. What was the 
height of the obelisk ? Ans. 57.62 feet. 

5. "Wanting to know the distance between two inaccessible 
trees, from the top of a tower 120 feet high, standing in a 
straight line with the two trees, I took the angles of depres- 
sion to the base of each tree, and found them to be 57°, and 
25° 30'. What was the distance between the two trees ? 

Ans. 173.656 feet. 



6. From a window near the foundation of a house, which 
seemed to be on a level with the base of a steeple, I took the 
angle of elevation of the top of the steeple, and found it to 
be 40°; then from another window, 18 feet directly above 
the former, the like angle was 37° 30'. What was the 
height, and what the distance, of the steeple ? 



A ns. 



210.44, and 250.79 feet. 



7. Wishing to know the length of a certain pond of water, 
I measured a line 100 yards in length, and at each of its 
extremities observed the angles subtended by the other 
extremity and a couple of trees at the extremities of the pond. 
These angles were, at one end of the line 32° and 98°, and 
at the other 37° and 118° ; what was the length of the 
pond ? 

Draw the horizontal line AB 
equal to 100 ; make the angle 
BAD 32°, BAG 98°, ABC 37°, 
and ABD 118°. The intersec- 
tions of the lines AC and BC, 
AD and BD determine the ex- 
tremities of the pond ; the straight line CD is the length of 
the pond. 




68 BOOK II. 

In the triangle ABC, subtract the sum of the angles at A 
and B from 180°, to find the angle ACB ; then find the side 
BC. 

In the triangle ABD, subtract the sum of the angles at A 
and B from 180°, to find the angle ADB ; then find the side 
BD. 

Subtract the angle ABC from ABD, to find the angle 
CBD ; then in the triangle CBD we shall have the two sides 
BC and BD, and the included angle CBD, to find the side 
CD. Ans. 161.868 yards. 

8. Being on the side of a river, and wanting to know the 
distance to a house on the other side, I measured 200 yards 
in a straight line by the side of the river ; and then at each 
end of this line took the horizontal angle between the house 
and the other end of the line. These angles were 68° 2' and 
73° 15' ; what were the distances from each end to the house? 

Ans. 296.54, and 306.19 yards. 

9. Having to find the height of a castle standing on an 
eminence, I measured the angle subtended by its base and 
summit, and found it to be 40° ; then measuring 40 yards in 
a straight line towards the castle, the like angle was 63° 20', 
while the angle of elevation of its summit, at the latter 
station, was 65°. What was the height of the castle ? 

Ans. 58.034 yards. 

10. Wanting to ascertain the breadth of a river, I measured 
a straight line 500 yards in length by one side of it, and at 
each end of this line I found the horizontal angles subtended 
by the other end, and a tree standing on the opposite margin 
of the river to be 53° and 79° 12'. What was the perpen- 
dicular breadth of the river ? Ans. 529.485 yards. 

11. Three objects A, B, and C, in a straight line, whose 
distances asunder were, AB 3.626, and BC 8.374 miles, were 
visible from one station, D, at which the angle ADB was 19°, 
and BDC 25°. Required the several distances DA, DB, and 
DC. 



PLANE TRIGONOMETRY. 



69 




On an indefinite straight line lay 
off AB equal to 3.626, and BC equal 
to 8.374. Make the angle ACF 19°, 
and CAF 25°. Describe a circle about 
the triangle ACF (Geom. 284) ; draw 
the straight line FB, and produce it 
to meet the circumference in D ; draw 
the straight lines DA and DC ; then 
the lines DA, DB and DC are the re- 
quired distances. 

For the angle ADB is 19°, being equal to ACF in the same 
segment ; and BDO is 25°, being equal to CAF in the same 
segment (Geom. 122). 

In the triangle ACF, the side AC is equal to AB+BC=12, 
and the two adjacent angles are given. Find the sides AF 
and FC. 

In the triangle ABF, find the angle ABF, which is equal 
to DBC; and 180° -the angle ABF gives the angle ABD. 

In the triangle ADB, find the sides AD and DB ; in the 
triangle DBC find the side DC. 

Ans. DA 9.47, DB 10.86, DC 16.85 miles. 

12. In a triangle, ABC, the side AB is 800, AC 600, and 
BC 400 yards. At a point, P, without the triangle, and on 
the opposite side of AB from the point C, the angle APC is 
33° 45', and BPC is 22° 30',— these angles being in the same 
plane with the triangle ABC ; what then are the distances 
PA, PB, and PC? 

With the given sides construct the triangle ABC. Make 
the angle ABF equal to the given angle APC; and the 
angle BAF equal to the given angle BPC. 

Describe a circle about the triangle ABF ; draw the 
straight line CF, and produce to meet the circumference on 
the opposite side ; this will determine the point P, — as will 
be seen by joining AP and BP (Geom. 122). 

Ans. PA 710.19, PB 934.29, PC 1042.52 yards. 



70 BOOK II. 



AEEAS OF TRIANGLES AND OTHER POLYGONS. 

(64.) The Area of any figure is its quantity of surface ; 
and is expressed by the number of times that its surface 
contains some assumed unit of surface, as a square inch, or a 
square foot, &c. 

1. The area of a Parallelogram is equal to the product of 
its base multiplied by its altitude (Geom. 205). 

2. The area of a Triangle is equal to half the product of 
its base multiplied by its altitude (Geom. 206). 

3. The area of a Trapezoid is equal to half the product of 
the sum of its two parallel sides multiplied by its altitude 
(Geom. 207). 

Geometry establishes the principles upon which areas are 
to be computed from the linear dimensions of the figures. It 
is sometimes most convenient to compute the area of a 
Polygon from determining lines and angles' this requires 
the aid of Trigonometry. 

(65.) To find the Area of any Parallelogram or Triangle 
from two Sides and the included Angle. 

Radius is to the sine of the given angle, as the Product of 
the two sides containing that angle is to the area of the 
Parallelogram, or double the area of the Triangle. 



Let ABCD be a parallelogram, 
id ABD a triangle, hs 
their common altitude. 



and ABD a triangle, having DP for 



The right-angled triangle AFD a. f 
gives 

Radius : AD ; : sin A : DF (M) ; and therefore 

__ sin A . AD 

DF=— — . 



PLANE TRIGONOMETRY. ft 

"We have therefore 

A ^ sin A . AD . AB ^ 

the area AC= — (Geom. 205). 

R 

Multiplying both sides of this equation by R, and con- 
verting the resulting equation into a proportion, we shall find 

E : sin A : : AD . AB : the area AG (Geom. 180). 

The parallelogram AC is double the triangle ABD ; hence 
Radius is to the sine of the given angle, as the product, &c. 

EXAMPLE. 

To find the area of a triangle, ABD, in which the side AB 
is 225 yards, AD 110 yards, and the included angle A 65 c 



o 



Radius 10 10 comp. log. 0.000000 

is to sin A 65° 9.957276 

AT? KT .\ 225 2,352183 

asAB.AD) no 2.041393 

is to cloy&le area 22431.2 4.350852 

In this operation the logarithm of the product 225x110 is 
obtained by taking the sum of the logarithms of the two 
factors (8...1), — this sum being added up with the preceding 
logarithms by adding each of its two parts. 

The required area is thus found to be 

22431.2^2=11215.6 square yards. 

1. Find the area of a Parallelogram in which two adjacent 
sides are 75.6 yards, and 120 yards, respectively, and their 
included angle 60° 25'. Am. 7889.36 square yards. 

2. Find the area of a Triangular piece of ground which has 
two of its sides respectively 240 rods and 325.8 rods in length, 
and their included angle 96° 30'. 

Ans. 38844.64 square rods. 

4* 



n 



BOOK II. 



3. Find the area of a Parallelogram in which an angle of 
87° 35' is included between two sides which are respectively 
340 feet and 578.3 feet in length. 

Ans. 196447.9 square feet. 

4. Find the area of a Triangular meadow which has two 
of its sides respectively 90 poles and 103.5 poles in length, 
and their included angle 100°. 

Ans 4586.74 square poles. 

(66.) To find the Area of a Triangle from two Angles 
and the included Side. 



The sum of the two given angles subtracted from 180° will 
leave the angle opposite to the given side ; then^ 

Radius into the sine of the angle opposite to the given side, 
is to the Product of the sines of the other two angles, as the 
square of the given side is to double the area of the Triangle. 



Suppose ABC to be a tri- 
angle in which the angles A 
and B, and the included side 
AB are given. 

Draw CD perpendicular to 
AB ; then in the two triangles 
ABC and ADC we shall have 




Sin ACB : sin B : : AB : AC (48) ; 
Radius : sin A : : AC : CD (44). 

Multiply together the corresponding terms of these pro- 
portions, 

R . sin ACB : sin A . sin B : : AB : CD (Geom. 194 
and 185) ; : AB 2 : AB . CD (Geom. 184). 

But AB . CD is double the area of the triangle ABC ; 
therefore, Radius into the sine of the angle opposite to the 
given side, &c. 



PLANE TRIGONOMETRY. 



73 



EXAMPLE. 



To find the area of a triangle, ABO, supposing the angle 
A to be 38°, the angle B 55°, and the included side AB 40 
poles. 

The angle ACB, opposite to the given side AB is 



180°-(38 o +55 o )=87°. 



E . sin 87° ... j " 

{OQO 




as AB 2 

is to double area 



40 
40 

808.02 



comp 
comp 



log. 0.000000 
log. 0.000596 
9.789342 
9.913365 

1.602060 
1.602060 

2.907423 



In this operation the logarithm of each of the three pro- 
ducts is obtained by taking the sum of the logarithms of the 
factors (8...1) ; these sums being added up by adding their 
constituent parts. The divisor, or first term of the propor- 
tion, being the product of two factors, we use the complements 
of the logarithms, and reject 20 from the sum of all the 
logarithms. 

The required area is thus found to be 

808.02-r-2=404.01 square poles. 

5. Find the area of a Triangle in which two of the angles 
are 80° and 60° respectively, and the included side 32 feet. 

Ans. 679.33 square feet. 

6. Find the area of a Triangular field having one of its 
sides 45 poles in length, and the two adjacent angles, respec- 
tively, 70° and 69° 40'. Ans. 1378.411 square poles. 

7. Find the area of a Triangular piece of ground having 
two angles respectively 73° 10' and 90° 50', and the side 
opposite to the latter 75.3 poles. 

Ans. 748.03 square poles. 




74 BOOK II. 

(67.) To find the Area of any Quadrilateral from its two 
Diagonals and the angles at their intersection. 

Radius is to the Sine of any one of the angles at the inter- 
section of the two diagonals, o)s the Product of the two 
diagonals is to double the area of the Quadrilateral. 

Let ABCD be any quadrilateral 
whose diagonals AC and BD intersect 
each other at F. 

The four angles at F haye equal 
sines, since the opposite angles are jt 
equal, and the adjacent angles' are 
supplements of each other (20). We. have therefore 

E : sin F : : AF . FB : 2 area AFB ; 
B : sin F : : AF . FD : 2 area AFD ; 
E : sin F : : FG . FB : 2 area FBG ; 
E : sin F ; : FC . FD : 2 area FCD (65). 

By converting each of these proportions into an equation 
(Geoin. 176), adding the equations together, and substituting 
equivalents, we shall find 

E. 2 area ABCD-sin F (AF+FC) (FB-fFD)=sin F . AG . DB 
which gives E : sin F : ; AC . DB : 2 area ABCD. 

Therefore, Eadius is to the sine of any one of the angles at 
the intersection, &c. 

8. Find the area of a Quadrilateral whose diagonals are 
36 feet and 75 feet in length, and intersect each other at an 
angle of 50° 25'. Ans. 1040.44 square feet. 

9. Find the area of a Quadrilateral piece of ground whose 
diagonals are 80 poles and 100 poles in length, and intersect 
each other at an angle of 85° 15'. 

Ans. 3986.26 square poles. 

10. Find the contents of a Quadrilateral tract of land whose 
diagonals are 75.5 poles and 130.6 poles in length, and inter- 
sect each other at right angles. 

Ans. 4930.14 square poles. 



PLANE TRIGONOMETRY. 



7o 



(68.) To find the Area of a Regular Polygon from one of 
its equal Sides. 

One of the equal angles of a regular polygon will be found 
by multiplying 180° by the number of sides of the polygon, 
subtracting 360° from the product, and dividing the remainder 
by the number of sides (Geom. 78). 

To compute the apothem of the polygon (Geom. 132). 

Eadius is to half the side of the polygon, as the tangent of 
half the angle of the polygon is to the apothem (46). 

The area of the polygon will then be found by taking half 
the product of the perimeter and apothem (Geom. 252). 



In the regular hexagon whose cen- 
tre is C, draw CD perpendicular to 
the side AB ; then CD is the apothem, 
AD is half of AB, and the angle 
DAC is half the angle of the hexa- 




The angle of a regular hexagon is 

(180 o x6-360°)-r6=120°. 

Hence the angle DAC is 60° ; and consequently 

Eadius : AD : : tang 60° : CD (46). 

The area of the regular Hexagon is expressed by 6 
CD or 3 AB . CD. 



AB.i 



11. Find the area of a regular Hexagon, and also of a 
regular Octagon, whose sides are each 10 feet. 

Ans. 259.8 ; and 482.84 square feet. 

12. Eind the area of a regular Pentagon, and also of a 
regular Decagon, whose sides are each 12 feet. 

Ans. 247.74; and 1107.96 square feet. 




76 BOOK II. 

(69.) To find the Area of any Polygon from its Sides 
and Angles. 

Divide the Polygon into triangles by drawing diagonals 
frqm one of its vertices. From the given sides and angles 
compute such parts of each triangle as will determine its 
area ; and take the sum of the areas of the several triangles 
for the area of the polygon. 

Let ABCDF be any pentagon 
whose sides and angles are given. 
Draw the diagonals AC and AD. 

In the triangles ABC and AFD the 
sides containing the angles B and F, 
and these angles themselves are 
given ; hence the areas of these two 
triangles may be computed (65). 

In the triangle ABC find the angle ACB and the side AC 
(57) ; subtract ACB from the given angle BCD, and the 
remainder will be the angle ACD. Then in the triangle 
ACD the sides AC and CD, and the included angle ACD 
will N be known ;" whence the area of this triangle may be 
computed. The sum of the areas of the three triangles is 
the area ABCDF. 

In the preceding computations no use would be made of 
the angles BAF and CDF. Moreover, by computing the side 
AD, in the triangle ACD (57), we should have the three sides 
of the triangle ADF — from which the area ADF might be 
computed ; and thus the angle AFD might also be dispensed 
with. 

It is also apparent that these three angles might be found 
from the sides and the other angles of the polygon. This is 
a particular case of the general truth, that 

(•70.) Any three parts of a Polygon, except three sides, 
may be computed, by Plane Trigonometry, when the other 
parts of the polygon are given. 



PLANE TRIGONOMETRY. 77 

This principle results from the polygon's being composed 
of triangles, and a triangle's being determined notwith- 
standing the want of any three of its parts, except the three 
sides (53). , 

A polygon may always be constructed from given parts 
determining the figure. The bases and altitudes of its con- 
stituent triangles might then be measured, instrumentally, 
in the figure (59), and an approximate area be thence readily 
computed. 

The computation of the areas of Polygons from their sides 
and angles, on the principles of mere Trigonometry, becomes 
more tedious as the number of sides is increased. A more 
practicable method — involving the angles which the sides of 
the polygon make with a system of parallels drawn through 
its vertices — will be explained under the head of Surveying, 
which comes next in the present course of mathematical 
studies. 



BOOK III. 

SURVEYING. 

(VI.) Surveying consists in the measurement and delinea- 
tion of any portion of the Earth's surface, regarded as 
uniformly level. 

A level surface is one which conforms to the general curv- 
ature of the Earth, — being such as an expanse of water 
presents, when perfectly at rest. A level surface is therefore 
not a plane surface, but nearly spherical ; the entire Earth 
itself being very nearly spherical, with a diameter of about 
7912 miles. 

From the diameter of the Earth we may determine the 

Deviation of a Level Surface from a Plane. 

(V2.) A level surface on the Earth, for moderate distances, 
deviates from a tangent plane proportionably to the square 
of the distance from the point of contact; and this deviation 
amounts to about 8 inches at the distance of one mile. 

Let A be the point in which a plane 
ip tangent to the surface of the Earth, 
and AB a straight line in that plane. 
Let C be centre of the Earth. Draw 
the straight line CB, cutting the cir- 
cumference of a great circle in D, and 
let DCF be the diameter of that circle. 
Then 

BD : AB ; : AB : BF (Geom. 234) ; 
AB 2 AB 2 




which gives BD 



BF FD+DB 



SURVEYING. 79 

For moderate distances, AB may be considered equal to 
AD ; and no important error can result from the omission of 
DB in the preceding division, since DB will be very small 
in comparison with the diameter of the Earth. We have 
therefore 

FD 

The diameter FD being a constant quantity, the deviation 
BD of the level surface from the tangent AB will vary as 
AD 2 , that is, it will be proportional to the square of the 
distance from the point of contact. 

If the distance AD be one mile, or 5280 feet, and the 
diameter FD of the Earth be taken at 7912 miles, we shall 
have 

BD= mile=.667 feet— 8 inches, nearly. 

7912 J J 

The deviation of the Earth's surface, when perfectly level, 
from a tangent plane, is thus found to be 8 inches for the 
distance of one mile / and this deviation having been shown 
to be proportional to the square of the distance,' we find 

The deviation at 2 miles to be 8x2 2 = 32 inches; 
" at 3 miles " 8x 3 2 = 72 inches ; 

" at 4 miles " 8xl 2 =128 inches ; 

" at 5 miles " 8x5 2 =200 inches, &c. 



PLANE SURVEYING. 

(V3.) Plane Surveying consists in the methods adopted 
for measuring the smaller divisions of the Earth's surface, as 
farms, townships, &c. ; in which the level surface is, without 
practical error, regarded as a horizontal plane. 

Geodesic Surveying (to be noticed hereafter) embraces 
larger portions of the Earth, and proceeds upon the consider- 
ation of its spherical curvature. 



80 BOOK III. 



Measurement of Distances. 

(74.) Linear measures in Surveying are commonly taken 
with a Chain, called, from its inventor, Gunter's Chain, 
which is 4 poles or 66 feet in length, and contains 100 "links ; 
but it is sometimes more convenient to use a two-pole Chain, 
containing 50 links. Each link in these chains is 

(66X12)H-100=7.92 inches long. 

It. is the horizontal or level .„.c%aiTv 

distance that must be measur- *fe^JL^ll§« 

ed in Surveying. On sloping g.„(%iakk L^f§ 

ground the chain must there- L^ggf 

fore be stretched horizontally, 

with one end held on the sloping surface ; a plumb-line, dh, 

held at the other end, will project the horizontal reach of the 

chain on the sloping surface. 

Thus the oblique line AB, measured horizontally, as repre- 
sented in the figure, is 3 chains, equal to the horizontal line 
AC, intercepted between the point A and the vertical line 
BO. 

The horizontal distance, AC, might also be computed, by 
Trigonometry, when the oblique distance AB and the angle 
BAC which the sloping surface makes with the horizontal 
base have been measured. Thus 

Radius : AB ; ; cos BAC : AC (44). 

By regarding the Radius as unity, we find the horizontal 
distance, AC, equal to the oblique distance, AB, multiplied 
by the natural cosine of the angle, BAC, which the surface 
makes with the horizontal base (45). 

Expresssion of Areas. 

(75.) The Area of a tract of land is the quantity of surface, 
in horizontal extent, included within its bounding lines ; and 
is usually expressed in acres, roods, said perches. 



SURVEYING. 81 

30^ square yards make 1 perch or square rod ; 
40 perches " 1 rood ; 

4 roods " 1 acre. 

Also 16 perches or square rods make 1 square chain ; 
10 square chains make 1 acre. 

The contents of the tract surveyed will be found in hori- 
zontal extent when the bounding lines have been measured 
horizontally. 

It would be extremely difficult to compute the actual, 
uneven surface of the ground ; and it is only by taking the 
horizontal or level extent that the different portions of the 
tract surveyed can be delineated, in plots or maps, in their 
true positions with respect to each other. Moreover, the value 
of land, for its productiveness and for other uses, is, in 
general, proportionate to its level extent, and not to its uneven 
surface. It is therefore an ordinance of law, as well as a 
convenience in practical science, that " Every survey shall 
be made by horizontal measurement." 

Meridians. 

(76.) Meridians of the Earth are great circles which cross 
the equator at right angles, and meet each other at the north 
and south poles ; but, for the comparatively small distances 
embraced in Plane Surveying, these meridians may be 
regarded as straight lines parallel to one another. 

The Magnetic Meridian of any place is the direction in 
which the magnetic needle, when freely movable on a pivot, 
rests at that place. This direction does not generally coincide 
with the true meridian passing through the poles of the 
Earth ; but the magnetic meridians are also parallel to one 
another to the extent of small surveys. 



Bearing of a Line. 

(77.) The Bearing or course of a line is the angle which it 
makes with a ineridian passing through the point from which 



82 



BOOK III. 



the line proceeds ; and is estimated from north or south 
towards east or west. 



Let !N"S be a meridian, or north and south line 
passing through the point A from which the 
line AB proceeds ; then the angle EAB is the 
hearing of the line AB. 

The top of the map or plot being north, the 
bottom is south, the right-hand side east, and 
the left-hand side west. 




B 



The line AB therefore runs towards the north and east 



and supposing the angle NAB to be 32 c 
line AB is 

K 32° E. ; read north 32° 



the bearing of the 



east. 



The Reverse Bearing of a line is its bearing from the other 
end of the line. Thus the reverse bearing of AB is the 
bearing of BA, that is, the angle S'BA. 

The bearing and the reverse bearing of a line are alternate 
angles which that line makes with parallel meridians, and 
are therefore equal to each other (Geom. 26), but are between 
opposite points. Thus the line BA runs towards the south 
and west, so that the reverse bearing of AB is 

S. 32° W. : read south 32° west. 



When the line makes an angle of 90° with the meridian, 
it then runs due east, or west. Thus the bearing of the line 
the bearing of BD is W., west. 



AC is E., east j 



Measurement of Bearings. 

(■y§.) The Bearings of lines, in Surveying, are usually 
measured with the Compass — in which a magnetic needle, 
settling freely on a pivot, indicates the meridian and the 
angle it makes with any straight line in which the sights of 
the instrument are directed 



SURVEYING. 



83 




The Compass is set on a staff or tripod, with its graduated 
circle in a horizontal position — as shown by spirit-levels, and 
its centre vertically above the point at which the bearing is 
to be taken. 

The sights of the Compass consists of two slits in the stan- 
dards A and B, perpendicular to the circle. These sights 
are directed to a staff set up vertically at some distance on 
the line whose bearing is to be measured. 

The letters E and "W", east and west, are reversed from their 
natural position on the face of the instrument. This is for 
convenience in reading the bearings ; thus, the observer being 
at the South end of the instrument, the sights will be directed 
~N. W. when the North end of the needle is nearest to W. 

The Compass-circle is usually graduated into degrees and 
half-degrees ; quarter-degrees are then readily estimated by 
the eye. A Vernier is sometimes attached which measures 
minutes (6 3... 2). 

The Vernier is represented on the right of the graduated 
circle, in the preceding figure. It is firmly attached to this 
circle, and is moved concentrically with it, through a few 
degrees, by turning a screw. On the right of the Yernier is 
a graduated arc, which marks the movements of the former 
in minutes. 

When the Bearing of a line is desired to minutes, let the 
of the vernier first coincide with the of the graduated arc. 
After the sights have been directed, and the needle has 
settled, move the circle, by turning the screw, until the line 



84 BOOK III. 

which marks the whole degrees in the bearing is brought to 
the point of the needle ; the vernier will then show the 
number of minutes to be added. 



Forward and Back Sights. 

(79.) The Bearing of a line and its reverse bearing are 
equal angles (77). The reverse bearing should be taken, by 
a back sight, at the termination of every line, as a test of 
correctness. 

"When the forward and the .bach sight give equal angles, 
the bearing may be regarded as correctly measured ; when 
unequal, the bearing should be taken again, with greater 
care. 

When these two angles cannot he found equal, the needle 
has been deflected from its proper direction, at one of the 
two stations, by the attraction of some ferruginous substance, 
concealed, it may be, in the earth. 

To determine at which station the deflecting cause exists, 
choose a third station, and compare the bearing and the 
reverse bearing between this station and each of the other two. 
This will generally show whether the deflection was at the 
first or second station, and, consequently, whether the bearing 
or the reverse bearing is the proper angle. The difference 
of these angles will show the amount of the needle's deflection, 
east or west, and will be a correction to be made in any other 
bearing indicated by the Compass at the deflecting station. 

Difference of Latitude and Departure. 

(§0.) The Difference of Latitude, or the northing or south- 
ing, of a line, is the distance that one end of it is north, or 
south, of the other, — being the part of a meridian intercepted 
between one end of the line and a perpendicular drawn from 
the other end. 

Thus AD is the northing of the line AB ; and BC is the 
southing of the line B A. 



SURVEYING. 

Tke Departure, or the easting or westing, of a 
line, is the distance that one end of it is east, or 
west, of the other, — being the perpendicular 
distance between the meridians passing through 
the extremities of the line. 

Thus, DB is the easting of the line AB ; and 
CA is the westing of the line BA. 



jST 



(S 



85 

3 

a 



(g|P For brevity, we shall, in this treatise, use the simple 
term Latitude instead of "Difference of Latitude." Thus, 
AD is the latitude of the line AB. 



Computation of Latitudes and Departures. 

(§1.) The hearing of a straight line is one of the acute 
angles, the line itself, commonly called the distance, is the 
hypothenuse, and its latitude and departure are the perpen- 
dicular sides, of a right-angled triangle. 

If any two of these quantities be given, the other two may 
be computed by Trigonometry. It is generally required to 
find the latitude and departure when the bearing and distance 
are given. 



In the right-angled triangle ADB, the 
angle A is the hearing of the line or distance 
AB, AD is the latitude, and DB the de- 
parture of AB. 

R : AB ; ; sin A : DB, or cos A : AD (M); 

that is, Radius is to the distance, as the sine 
of the bearing is to the departure, or as the 
cosine of the bearing is to the latitude. 



If the Radius be unity, we shall find 

DB=ABxnat. sine A ; AD=ABxnat. cos A (45). 
Hence, the latitude will be found by multiplying the 




86 BOOK III. 

distance by the natural cosine, and the departure by multi- 
plying the distance by the natural sine, of the bearing. 

In this way the latitudes and departures corresponding to 
any bearings and distances may be computed. Such com- 
putations have furnished the contents of the 



TE AVERSE TABLE. 

(§2.) Table III., appended to this work, contains the lati- 
tudes and departures for bearings expressed in degrees and 
fourths, and for distances from 1 to 10 ; the fourth of a degree 
being the smallest angular magnitude estimated in Common 
Surveying, and the distances, in this Table, being easily 
extended by addition. 

Find the bearing, when less than 45°, in one of the left- 
hand columns of the Table ; find the distance, if not more 
than 10, at the top of the page ; then opposite to the bearing, 
and under the distance, will be found the Lat. and Dep. as 
marked at the top. 

Thus, for bearing 20J°, and distance 4, we find 

Lat. 3.7528 ; Dep. 1.3845. 

Find the bearing, when more than 45°, in one of the right- 
hand columns of the Table ; find the distance, if not more 
than 10, at the bottom of the page ; then opposite to the 
bearing, and over the distance, will be found the Lat. and 
Dep., as marked at the bottom. 

Thus, for bearing 70J°, and distance 9, we find 

Lat. 3.0413 ; Dep. 8.4706. 



(§3.) To find from the Traverse Table the Latitudes and 
Departures for Distances exceeding 10. 

1. Find the Lat. and Dep. for each numeral figure of the 
given distance, and remove their decimal points one figure 
to the right or left for each place the figure in the distance 
is removed to the left or right of units. 



SURVEYING. 



87 



2. Take the sums of these partial latitudes and departures 
for the Lat. and Dep. corresponding to the given bearing and 
distance. 

Thus, to find the Lat. and Dep. of a line which runs 
K 25° E?, 345.67 chains. 

We find the Lat. and Dep. for the distances 3, 4, 5, 6, 7, 
separately. For the 3 we remove the decimal point two 
figures to the right, because the 3 is removed two places to 
the left of units ; for the 4 we remove the point one figure to 
the right / for the 6 we remove the point one figure to the 
left, because the 6 is removed one place to the right of units ; 
for the 7 we remove the point two figures to the left. 

The results will be — using no more than four decimals : 



Lat. for 300 .... . 271.89 ; Dep. 

" " 40 36.252 ; " 

" " 5 4.5315; " 

" " .6 5437 ; " 

" " .07 . . . _ .0634 ; " 

Lat. for 345.67 . . . 313.2806 ; Dep. 



126.79 

16.905 

1.1131 

.2535 

.0295 

145.0911 



By the preceding method the latitudes and departures, for 
the same bearing, are increased, or diminished, proportion ably 
to the distance ; the geometrical principle involved being 
that of the proportionality of the sides of similar triangles. 

For decimal figures in the Distance, the Table will give 
more than four decimals in the Lat. and Dep. It will be 
sufficiently accurate to use only the first four decimals, which 
we shall never exceed in taking the numbers from the Table. 

Differences of Latitude are to be designated as north or 
south, Departures as east or west, according to the bearings 
of the lines. 



Find, from the Traverse Table, the latitudes and departures 
for the following bearings and distances. 

5 



gg BOOK 111. 

1. K 151° E., 9 chains. 

Ans. N. 8.6831; E. 2.3673. 

2. S. 83£° E., 10 chains. 

Ans. S. 1.1320 ; E. 9.9357. 

* 

3. K 31J° W., 234 chains. 

Ans. K 198.9824; W. 123.1309. 

4. S. 701° "W., 408.5 chains. 

Ans. S. 136.3574; W. 385.0724. 

5. K 89i° E., 560.25 chains. 

Ans. K 7.3282; E. 560.2048. 

6. S. 40° E., 175.3 chains. 

Ans. S. 134.283; E. 112.6817. 

7. K 53J° W., 300 chains. 

Ans. K 179.50; W. 240.38. 

8. B. 13|° W., 230.5 chains. 

Ans. S. 223.8956; W. 54.7898. 



Field Notes of a Survey. 

(84.) The Eield Notes are the surveyor's memorandum or 
record of the angles and distances measured on the ground, 
and of any other particulars which may be useful towards a 
correct delineation, description, or computation of the survey. 

The surveyor usually begins at one corner of the field or 
tract of land, and, going entirely around it, measures the 
hearing of each side with the Compass ; his assistants follow, 
and measure each side, horizontally, with the Chain, or some 
other suitable instrument. 

When the bearings of the sides have been thus measured, 
it is sometimes desirable to find the angles contained 
between contiguous sides ; that is, to determine, by means of 
the Compass, the horizontal angle contained between any 
two horizontal lines which meet each other. 



SURVEYING, 



89 



(85.) To find the Angle contained between two Straight Lines, 
when the Bearings of those lines from the same point 
are given. 



1. When the Bearings are both Worth or South, — subtract 
the less from the greater if loth are east or west, but add 
them together if one is east and the other west. 

2. When one of the Bearings is North and the other South, 
— subtract their sum from 180° if loth are east or west, but 
subtract their difference from 180° if one is east and the other 
west. 

The correctness of these rules will readily appear from a 
diagram representing the different bearings of the lines. 

Let AB and AC be two straight 
lines, AB running south and west, 
AC north and east from the point 
A. 

Produce CA in a straight line 
to D. The angle DAS is equal 
to the vertical angle NAC, so that 
DAB is the difference of the lear- 
ings SAB and NAC ; and DAB 
subtracted from 180° leaves the 
angle BAC contained between the two lines AB and AC. 

In applying the preceding rules to two consecutive sides 
of a survey, one of the two bearings must be reversed, in order 
that both of them may be from the same point. 




Suppose FG to run K 50° E., and GK 
to run S. 35° E. The reverse bearing of 
FG, that is, the bearing of GF, is S. 50° W. 

The bearings of the two lines, taken from 
the same point G are therefore S. 50° W. 
and S. 35° E. ; and, according to the first 
of the preceeding rules, the angle FGK is 



50°+35°=85 c 



1ST 




90 



BOOK III. 



Plotting a Survey. 




(§6.) Plotting a Survey consists in constructing a plane 
figure which shall represent the tract surveyed according to 
the relative positions and lengths of its bounding lines. 

The more usual method of exe- •$ ^ 

cuting this is shown by the fol- 
lowing Example. 

To plot a survey of which the 
Field Notes are 

1. K 83° E., 11 chains ; 

2. S. 14° E., 23 chains; 

3. K 77° "W., 23.66 ch. ; 

4. K 23° E., 17 chains. 

Draw any straight line, USTS, 
for a meridian. At any point 
A in this meridian make the 

angle NAB, between north and east, 83° (59. ..3 or 4) ; and 
the line AB 11 (59.. .2). Through the point B draw the 
meridian 1ST' S', parallel to M> ; make the angle S'BC, 
between south and east, 14°, and the line BC 23. 

In like manner, draw the sides CD and DA, according to 
their given bearings, or the angles which they make with the 
meridians passing through the points and D, making these 
sides 23.66 and 17, respectively. 

"When the Field Notes have been correctly taken, and 
extend entirely around the tract surveyed, the Plot should 
close, or come together, at the point of beginning. Thus, the 
last side DA should terminate at the point from which the 
first side AB was drawn. 

If the angles contained between each two consecutive sides 
of the survey be found from the given hearings (85), the Plot 
may then be drawn with a single meridian. 

Thus the angle ABC will be found to be 97°, from which 



the side BC may be drawn, without the meridian N' S'. 
side CD may be drawn from the angle BCD, &c. 



The 



SURVEYING. 91 

The angle BCD will be found to be 63°, CD A 80°, and 
DxVB 120°. 



1. 


H" 


2. 


s. 


3. 


K 


4. 


S. 


5. 


s. 



The angle contained between two consecutive sides 
may be a re-entrant angle of the tract surveyed (Geom. 82) ; 
and this circumstance must be attended to in plotting by this 
last method. 

If the Surveyor has the tract on his right, in taking the 
bearings, then any side which runs to the left of the pre- 
ceding course will form a re-entrant angle with the preceding 
side ; and vice versa. 

By each of the two preceding methods plot the survey of 
which the Field Notes are as follows : 

34° E., 46.10 chains ; 
70° E., 8.00 chains ; 
84° E., 18.40 chains; 
8i° W., 32.26 chains ; 
84f° W., 46.44 chains; 
closing at the beginning. 

An attentive consideration of these bearings will show that 
the tract lay on the right of the Surveyor, as he passed around 
it, and that the 3d side forms a re-entrant angle with the 2d. 

Balancing the Latitudes and Departures. 

(87.) "When the Bearings and Distances have been correctly 
measured entirely around any tract of land, and the cor- 
responding latitudes and departures computed, or found from 
the Traverse Table ; it is evident that the sum of the northings 
should be equal to that of the southings, and the sum of the 
eastings equal to that of the westings. But 

From unavoidable inaccuracy in the use of the instruments, 
these sums will seldom or never exactly balance each other. 
"When the differences are so small as to be attributable to 
unavoidable errors in the Field Notes, the Latitudes and 
Departures may be balanced by applying to them the cor- 
rections found by the following proportions : 



92 



BOOK III. 



The sum of all the sides is to the length of any side, as the 
whole error in Latitude or Departure is to the collection in 
the latitude or departure of that side. 



In these proportions it will be sufficient to set down 
the first two terms in the nearest integers — omitting the 
decimals : this will shorten the work. 



Scholium. — The amount of error that may be allowed in 
the Latitudes or Departures, without resorting to a re-survey 
for its correction, is a point on which surveyors differ. The 
best general criterion that can be given for it appears to be, 
that the difference between the sums of the northings and 
southings, or of the eastings and westings, should not exceed 
1-SOOth of the sum of all the sides of the survey, which is an 
error of 1 link for every 3 chains. 

. EXAMPLE. 



1 


COUESES. 


DIST. 


N. 


S. 


B. 


"W. 


C. 1. 


c. d. 


, 


S. 


E. 


"W. 


N. 36° E. 


125 


101.13 




73.47 




.05 


.0! 


101.18 




73.46 




i 2 


North 


54 


54.00 








.02 


.00 


54.02 








3 


S. 81° E. 


186 




29.08 


183.71 




.07 


.01 




29.01 


1S3.70 




1 4 


S. 8° W. 


137 




135.65 




19.20 


.05 


.01 




135.60 




19.21 


| 5 


West 


130 








130.00 


.05 


.01 


.05 






130.01 


6 


S. 40° W. 


70 




53.62 




45.00 


.02 


.00 




53.60 


45.00 


7 


N. 45° W. 


89 


62.93 






62.93 


.03 


.01 


62.96 


| 


62.94 




791 


218.06 


218.35 
218.06 


257.18257.13 
257.18| 


.29 


.05 


218.21 


218.2l! 257.16 

1 


257.16 






.29 


.05 

















The second and third columns of this Table contain the 
courses and lengths of the sides. The next four columns, 
marked N. S. E. W., contain the corresponding latitudes and 
departures. 

Observe that the 2d side, running north, has its entire 
length for difference of latitude, and has no departure ; and 
that the 5th side, running west, has its entire length for 
departure, and has no difference of latitude. The latitudes 



SURVEYING. 93 

and departures of the other sides are found from the Traverse 
Table, with four decimal figures, the third and fourth of 
which are rejected, but the second is increased by 1 when 
the third is 5 or more / this gives the nearest hundredths, 
which is sufficiently accurate. Thus the northing of the first 
side, 125, was found to be 101.1251, which is set down as 
101.13. 

The sums of the northings and southings is found to differ 
by .29, and the sums of the eastings and westings by .05. 

To find the corrections to be made in these latitudes and 
departures. The sum of all the sides is 791 ; then, 

791 : 125 : : .29 : .015 ; 
791 : 125 : : .05 : .007. 

We have thus found .045 for the correction to be made in 
the latitude, and .007 for the correction to be made in the 
departure, of the first side. Taking the nearest value in 
hundredths, these corrections become .05 and .01, which are 
put in the columns marked c. I. and c. d., opposite to the first 
side. 

In like manner we find the other corrections in those, two 
columns. 

The sum of all the corrections in latitude must be equal to 
the whole error in latitude / and the sum of all the corrections 
in departure must be equal to the whole error in departure. 
When these equalities are not obtained, one or more of the 
corrections must be sufficiently increased or diminished. 

To apply the corrections : When latitude is in the greater of 
the two columns, the correction must be subtracted from it ; 
when it is in the less column, the correction must be added to 
it ; and the same rule must be followed for the departures. 

Thus in the preceding Table the sum of the northings is 
less than that of the southings, and the sum of the eastings is 
greater than that of the westings ; the correction .05 for the 
first latitude must therefore be added to the northing 101.13, 
and the correction .01 for the first departure must be sub- 
tracted from the easting 73.47. 



94 BOOK III. 

"We thus find the corrected latitudes and departures in the 
four right-hand columns of the Table. 

jjglP Observe that, as the 5th side has no difference of 
latitude, the correction .05 found for that side is put in the 
column of corrected northings, because it is the sum of the 
northings that needs to be increased. 

By adding up the last four columns, it is found that the 
sum of the northings is equal to that of the southings, and the 
sum of the eastings to that of the westings. 

By the preceding method the corrections for the latitudes 
and departures are distributed in proportion to the lengths of 
the several sides / and this may often be done, with sufficient 
accuracy, by a mere mental estimation of the required cor- 
rections. 

Thus, in the preceding Example, the error .05 in departure 
might have been at once distributed by assigning a correction 
of .01 for each of the five longest sides of the survey. 



A problem of not unfrequent occurrence in Surveying is 
that in which it is required 

(§§.) To find the Bearing and Length of one Side of a tract 
when the hearings and lengths of the other sides are given. 

1. The difference between the sums of the northings and 
southings of the given sides will be the latitude of the 
required side ; and the difference between the sums of their 
eastings and westings will be its departure. 

2. This latitude and departure will form the perpendicular 
sides of a right-angled triangle, in which find the angle 
opposite to the departure for the hearing, and the hypothenuse 
for the length, of the required side. 

For an Example we take the same Field I^otes as before, 
omitting the bearing and length of the 3d side ■•' with the 
latitudes and departures of all the other sides, as found from 
the Traverse Table. 



SUKVEYING 



95 





C0T7BSES. 


DIST. 


N. 


s. 


E. 


W. 


1 


N. 36° E. 


125 


101.13 




73.47 




2 


North 


54 


54.00 








3 














4 


S. 8° W. 


137 




135.65 




19.20 


5 


West 


130 








130.00 


6 


S. 40° W. 


70 




53.62 




45.00 


7 


N. 45° W. 


89 


62.93 






62.93 


218.06 
189.27 


189.27 


73.47 


257.13 
73.47 



S. 28.79 



E. 183.66 



The sum of the northings of the given sides exceeds the 
sum of the southings by 28.79 ; and the sum of the westings 
exceeds the sum of the eastings by 183.66. Then the 3d side 
has 

South latitude 28.79, and east departure 183.66 (87). 



In the right-angled triangle ACB, 
let AC be equal to 28.79, and CB 
equal to 183.66 ; then the angle 
CAB is the required hearing, and 
AB is the required side. 




AC 28.79 1.459242 

is to K 10 10 10.000000 

as CB 183.66 2.261015 

is to tang. CAB 81° 5' . . . 10.801773 (46.) 

Sine CAB 81° 5' 9.991719 

is to CB 183.66 2.261015 

as sine C 90° . 10.000000 



is to AB 185.97 . 2.269296 (48.) 

The angle CAB, that is, the bearing of the 3d side, meas- 
ured with the Compass, and given in the Field Notes of the 

survey, is 81°, and AB, the length of that side, is 186. The 

5* 



96 BOOK III. 

want of perfect agreement between the measured and the 
computed quantities is owing to unavoidable inaccuracies 
in the measurements made on the ground. As heretofore 
remarked, the Surveyor does not usually take account of 
angular quantities less than one fourth of a degree. 

When the bearings and lengths of all the sides of a tract 
of land are measured, the balancing of the latitudes and 
departures affords a test of the accuracy with which the 
survey has been made. This test cannot be applied if any 
one bearing or side be omitted. The Field Notes should 
therefore be made complete, in all cases in which it is 
practicable. 



AREA OF THE SURVEY. 

(89.) To find the Area of the Survey from the Latitudes 
and Departures of its sides. 

1. Set the latitudes and departures — corrected when 
necessary to make them balance (87) — in their appropriate 
places in the columns 

K S. E. W. 

2. Transfer to the right, and add to itself that easting 
which will admit of each of the following ones being twice 
added, and the westings twice subtracted, in the order in 
which they occur, quite* around the survey. 

3. Multiply the first number in each pair of sums, or 
remainders, thus obtained, by the corresponding northing, or 
southing ; set the products in corresponding columns of north 
or south areas, and take half the difference between the sums 
of those two columns for the area of the survey. 

The correctness of the sums and remainders found by 
adding the eastings and subtracting the westings, will be 
shown by the last remainder's being 0. 

It may also be remarked, that the numbers to be multiplied 
might be obtained by adding the westings, and subtracting 
the eastings, in the 2d part of the Kule. 



SURVEYING 



97 



EXAMPLE. 



1 


N. 


s. 


E. 


•w. 


MUXT. N. AREAS. 


8. AEEAS. 




25.64 


13.0*7 




j 75.03 ) 

( 88.10 j" 




1923.7692 


2 




8.94 




11.41 


( 76.69 [ 
{ 65.28 ) 




685.6086 


3 




18.32 


5.10 




j 70.38 ) 
( 75.48 f 




1289.3616 


4 








17.71 


j 57.77 ) 
] 40.06 f 






5 


9.50 






6.90 


j 33.16 I 
( 26.26 } 


315.0200 




6 


16.06 








j 26.26 ) 
| 26.26 \ 


421.7356 




7 


6.97 






13.13 


j 13.13 ) 
( 00.00 J 


91.5161 




8 


25.50 




8.80 


i 8 - 80 I 
!1 1*7.60 j" 


224.4000 




9 




5.13 


22.18 


j( 39.78 ) 

n 6i - 96 j 




204.0714 




58.03 


58.03 


49.15 


49.15 


1052.6717 


4102.8108 



4102.8108-1052.6717=3050.1391 ; 

then 3050.1391-^2=1525.0695 is the area. 

By adding up the columns of latitudes and departures, we 
find that the northings and southings balance, and also the 
eastings and westings. 

Upon considering the eastings and westings in their rela- 
tions to one another, we perceive that if we begin with 8.80, 
the subsequent eastings, 22.18, 13.07, &c, may be added, and 
the westings, 11.41, 17.71, &c, subtracted, in the order in 
which they occur, without having a larger subtrahend than 
minuend. 

We therefore transfer the 8.80 to the next column on the 
right, and add it to itself ; to the sum 17.60 we add the next 
easting 22.18 ; to the sum 39.78 we add the same easting 
again ; to the sum 61.96 we add the next easting 13.07 ; and 
to the sum 75.03 we add the same easting again. From the 
sum 88.10 we subtract the westing 11.41, and from the 
remainder 76.69 we subtract the same westing again. To 
the remainder 65.28 we add the easting 5.10 : and so on, 



98 BOOK III. 

adding each easting twice, and subtracting each westing 
twice. 

There being no easting nor westing for the 6th side, the last 
number 26.26 found against the 5th side is set against the 
6th side, without increase or diminution. — After twice sub- 
tracting the last westing, the remainder is 0, which proves 
the additions and subtractions to have been correctly per- 
formed. 

We now multiply the first of the two numbers which stand 
against each side of the survey by the corresponding northing 
or southing, and set the product, accordingly, in a column of 
north or south areas. 

Thus we multiply 75.03 by 25.64, and set the product 
1923.7692 in the column of south areas ; we next multiply 
76.69 by 8.94:, and so on. 

There being no northing nor southing for the 4th side, the 
number 57.77 has no multiplier ; that is, the multiplier is 0, 
and the product is therefore 0. 

Having added up each column of areas, we take half the 
difference of their sums for the area of the survey. The area 
is found in square units corresponding in denomination to the 
linear units in which the sides are measured. 

Thus if the sides are measured in chain*, the area will be 
found in square chains / and will then be reduced to acres 
by dividing it by 10 (75). 

Demonstration of the preceding Hide. 

Let N S be a meridian pass- ^ 
ing; through the most eastern 

! "T? 

point of the survey ; and sup- it """7! ^^^ 

pose the Field Notes to have J / j """"""""^-^ n 

i fy... ./ i _ # — ■Sj 

been taken around the tract "\ 7 ° 

in the direction ABCD, &c. ; a/ 

then the easting gB is the first TV 

multiplicand found accord- ^j ™h^ J'. - . 

ing to the Rule ; and \^ ./ 

gQ . A^=2 the area ABg #| r * 

(Geom. 206). si 



SURVEYING. 99 

The next multiplicand is gB+gB or Atf+the easting oC, 
equal to </B+AC; and 

{gB+hC) . Bo=2gBCh (Geom. 207). 

The next multiplicand is </B+hC+oG— the westing Cs, 
equal to ho+oC-\-hC— Gs, equal to AC+As or mD; and 

(AC+mD) . sT>=2hGDm. 

The next multiplicand is AC+mD— Cs— the westing Dr, 

equal to AC— Cs-+mD— Dr, equal to As or mD+^r or jpF ; 

and 

(mD+j?F) . r¥=2mD¥p. 

The next multiplicand is niD+pF— Dr— the westing Fj?, 
equal to mD— Dr+^F— Fj?, equal to mf or ~Fp; and 

Fp .^A=2AF^. 

JSow the area of ABCDFA is equal to the sum of the areas 
of the three preceding trapezoids diminished by the areas of 
the two triangles; and a like equality obtains with respect 
to the double areas. Observe that the trapezoids give south 
areas, since the multipliers Bo, sD, and rF are southings y 
while the triangles give north areas, since the multipliers Ag, 
andj?A are northings y so that half the difference between 
the sums of the north and south areas is the required area of 
ABCDFA. 

This demonstration sufficiently develops the principles of 
the Kule. 



Graphic Method of Computing the Area. 

(90.) The Plot or map of a survey, like any other Polygon, 
may be divided into triangles by diagonal lines ; the bases 
and altitudes of these triangles mav be measured on the same 
scale from which the sides of the Plot were taken (59. ..2), 
and approximate areas of the triangles be thence computed. 
The sum of these areas will be an approximate area of the 
survey. 



100 BOOK III. 

With a view to this method it may be remarked that the 
most accurate method of plotting a survey is by means of the 
balanced latitudes and departures of its sides. — For an 
example, we recur to the preceding figure. 

Draw a meridian JSTS, and upon it lay off Ag, equal to the 
northing of the side AB ; draw the perpendicular gB equal 
to the easting of the same side, and then from A to B draw 
the side AB. 

Again; on the same meridian lay off gh equal to the 
southing of BC ; draw the perpendicular AC equal to the sum 
of the eastings of AB and BC, and then from B to C draw 
the side BC. 

Again ; on the same meridian lay off hm equal to the 
southing of CD ; draw the perpendicular mD equal to the 
sum of the eastings of AB and BC minus the westing of CD, 
and then from C to D draw the side CD. 

Proceed in like manner with the remaining sides — accord- 
ing to the directions of their latitudes and departures — to the 
last one, which must be drawn to terminate in A, the point 
of beginning. 

The superior accuracy of this method of plotting consists 
in its being independent of the hearings of the sides, or the 
angles they make with each other — in the construction of 
which errors are most liable to occur. 

Scholium. — It is not necessary to draw a Plot of a survey 
in order to compute its Area from the Latitudes and Depart- 
ures (89), though for other purposes it will sometimes be 
necessary. When drawn by either of the two methods first 
described, it affords a test of the correctness with which the 
Field Notes have been taken, since the Plot must close, or 
nearly close, at the point of beginning, when the Bearings 
and Distances have been accurately measured. 

The student may be exercised in plotting from the Field 
Notes in the following 



SURVEYING. 101 

PROBLEMS 

In the Computation of the Areas of Surveys. 

1. Given the following bearings and distances, 

1. K 16J° E., 14.35 chains ; 

2. East, 7.82 chains ; 

3. S. 3J° TV., 14.45 chains ; 

4. 1ST. 86^° TV., 11.07 chains, 

to the point of beginning,— find the area. 

* Ans. 13 A., 1 E., 19.39 P. 

2. Given the following bearings and distances, 

1. 1ST. 27^° E., 9.45 chains ; • 

2. S. 63}° E., 8.28 chains ; 

3. S. 15i°E., 1.04 chains; 

4. S. 20 3 2 ° TV., 5.83 chains ; 

5. K 79f° TV., 10.15 chains, 

to the point of beginning,— to find the area. 

Ans. 7 A., 1 E., 36.36 P. 

3. Given the following bearings and distances, 

1. ]ST. 48° TV., 1.53 chains ; 

2. 1ST. 12|° TV., 1.12 chains ; 

3. K 77° TV., 1.64 chains ; 

4. S. 12i° W., 2.81 chains ; 

5. S. 76° E., 3.20 chains ; 
g > N. 24i° E., 1.14 chains, 

to the point of beginning,— to find the area. 

Ans. A., 3 E., 1.17 P. 

4. Given the following bearings and distances, 

1. 1ST. 84^° TV., 27.12 chains ; 

2. N. 4J° TV., 22.00 chains ; 

3. East, 16.58 chains ; 

4. North, 7.81 chains ; 

5. S. 76i E., 18.15 chains ; 

6. S. 10f TV., 28.42 chains, 

to the point of the beginning,— to find the area. 

Ans. 80 A., K., 29.27 P- 



102 BOOK III. 

5. Given the following bearings and distances, 

1. K45°¥, 89 chains; 

2. Wanting, Wanting ; 

3. North, 54 .chains ; 

4. S. 81° E., 186 chains; 

5. S. 8°W., 137 chains; 

6. West, 130 chains ; 

7. S. 40° W., 70 chains, 

to the point of beginning,— to find the bearing and distance 
of the 2d side (88), and the area of the survey. 

Am, 2d side K 35° 50' 33" E., 125.13 chains ; area 
. 3330 A., 1 R, 28.77 P. 

6. Given the following bearings and distances, 

1. North, 36.00 chains 

2. S. 84° W., 46.40 chains 

3. N. 53i° W., 46.40 chains 

4. Wanting, Wanting ; 

5. N. 22i° E., 56.00 chains 

6. S. 76J E., 48.00 chains 

7. S. 15° W., 43.40 chains 

8. S. 16f> W., 40.50 chains, 

to the point of beginning,— to plot the survey, find the bearing 
and distance of the 4th side, and compute the area. 

The Plot may be drawn from the given bearings and dist- 
ances thus: draw the first three sides according to their 
given bearings and distances; then, beginning at the same 
point as before, that is at the first station, draw the 8th, 7th, 
6th, and 5th sides according to their reverse hearings (77) and 
their given distances. Draw the 4th side from the termination 
of the 3d side to the beginning of the 5th, and the plot will 
be completed. 

Ans. 4th side S. 63° 32' 36", 42.50 chains; area 298 A., 
3 R, 33 P. 



SURVEYING. 103 

7. Given the following bearings and distances, 
1. S. 52° W., 58 poles; 



2. 


S. 15J Q E., 


76 poles ; 


3. 


West, 


70.9 poles ; 


4. 


K 36° W., 


47 poles ; 


5. 


North, 


64.3 poles ; 


6. 


K 621° W., 


59. poles ; 


7. 


K 19° E., 


108 poles ; 


8. 


S. 77° E., 


91 poles ; 


9. 


S. 27° E., 


115 poles, 



to the point of beginning, — to compute the area. 

Ans. 152 A., 1 K., 22.94 P. 

8. Given the following bearings and distances, 

1. K72f°E., 37.40 chains; 

2. S. 70f° E., 37.40 chains ; 

3. S. 53° W., 24.90 chains ; 

4. S. 83i° E., 48.20 chains ; 

5. S. 31i° W., 30.40 chains ; 

6. S. 62f ° W., 45.20 chains ; 

7. K 73i° W., 52.60 chains ; 

8. "Wanting, Wanting ; 

9. K 35§° E., 31.00 chains, 

to the point of beginning, — to plot the survey, find the bear- 
ing and distance of the 8th side, and compute the area. 

Ans. 8th side K 20° 31' 38" W., 30.31 chains; area 
579 A., 2 E., 20.01 P. 



(91.) To find the Area of a Survey, or of any Polygon, 
from the interior Angles and the Sides. 

1. Take for a meridian a straight line passing through the 
extremity of any side, at some assumed angle with that side, 
and consider that angle as the hearing of the side. 

2. Draw straight lines through all the other vertices of the 
polygon, parallel to the assumed meridian, and consider the 



104 



BOOK III. 



angles which these parallels make with the other sides as the 
"bearings of those sides. The area may then be found by the 
method of latitudes and departures (89). 



Let ABCDF be a tract of 
land of which all the sides and 
the angles ABC, BCD, &c, 
have been measured. 

Through the point A draw 
the straight line !N"S making 
any assumed angle, as 32°, 
with the side AB. 




Considering 1STS a meridian, the bearing of AB is N". 32° E. 
The bearing of BC is the angle ^BC, and will be found by 
subtracting AB^, which is equal to NAB (Geom. 26), from 
ABC. The bearing of CD is the angle DCA, and will be 
found by subtracting the sum of BCr (equal to </BC) and 
BCD from 180° (Geom. 20). The bearing of DF is the angle 
of FDs, and will be found by subtracting CD<? (equal to DC A) 
from CDF, and the remainder FD# from 180°. The bearing 
of FA is the angle raFA, equal to SAF, and will be found by 
subtracting the sum of FAB and BAN from 180°. 

Having the lengths of all the sides AB, BC, &c, and their 
bearings with respect to assumed meridians, the area may be 
computed by the Rule already given (89). 



This method of finding the Area does not require the use 
of the magnetic needle, since the angles ABC, BCD, &c, may 
be measured with any instrument which contains a graduated 
horizontal circle, with movable sights attached to it. 

In the Railroad Compass, of comparatively recent inven- 
tion, there is an arrangement for measuring horizontal angles, 
independently of the needle. "The accuracy and minute- 
ness of the horizontal angles indicated by this instrument, 
together with its perfect adaptation to all the purposes to 
which the "Vernier Compass can be applied, have brought it 



SURVEYING. 105 

into use in many localities, where the land is so valuable as 
to require more careful surveys than are practicable with a 
needle instrument." 

The sum of the interior angles of any polygon, when 
exactly measured, will be equal to twice as many right angles, 
wanting four, as the polygon has sides (Geom. 78). When 
there is a re-entrant angle, the number of degrees to be added 
for it, in finding the sum of the interior angles, will be 
obtained by subtracting the re-entrant angle from 360° 
(Geom. 82). 

When all the interior angles but one are given, that one 
may be found by subtracting the sum of the given angles 
from the proper number of right angles. 

(92.) When the hearing of one side and the interior angles 
of a survey are given, the hearings of the other sides may be 
found from the given angles, in the manner already explained 
for an assumed bearing of a side. 



The decimal parts of the latitudes and departures 
being, generally, but approximate values, the areas obtained 
from assuming different bearings for the first side of the 
survey (91...1), could not be expected to agree in all their 
decimal figures. In the 9th Problem the area is computed 
from assuming the side AB to run 1ST. 30° E. 

9. Find the area of a field whose angles and sides are as 
follows : 

the angle A 104°, the side AB 9.30 ch.; 
the angle B 105°, the side BC 11.85 ch.; 
the angle C 100°, the side CD 5.30 ch.; 
the angle D 130°, the side DF 10.90 ch.; 
the angle F 101°, the side FA 9.40 ch.; 

Am. UJl., 1 E., 39.2 P. 



106 BOOK III. 

10. The bearing of one side, AB, of a survey is 1ST. 30° E., 
and the remaining Field Notes are as follows : 

A re-entrant angle at A, 100°, the side AB 23.52 oh.; 

the angle B 58°, the side BC 30.55 oh.; 

the angle C 90°, the side CD 26.34 ch.j 

the angle D 100^°, the side DF 32.26 ch.; 

the angle F wanting, the side FA 18.35 chains. 

Find the bearings of the other sides, and the area of the 
survey. 

Ans. BO, S. 88° W. ; CD, S. 2° E. ; DF, 81J° E. ; FA, 
K 50° W. ; Area 69 A., 2 R, 39.3 P. 



SURVEYING BY OFFSETS. 

(93.) When the tract to be surveyed has an irregular side,' 
as the margin of a meandering stream of water, the following 
method may be adopted : — 

1. Run a straight line, called a station line, near to the 
irregular side, and so as to close with the rectilinear sides. 

2. Measure the distances on this line to a succession of 
perpendiculars, or offsets, drawn from it to the principal 
flexures in the irregular side; also measure those perpen- 
diculars. 

3. Compute the areas of the triangles or trapezoids thus 
formed between the station line. and the irregular side. Add 
these areas to the area inclosed by the station line and the 
rectilinear sides, or subtract them therefrom, according as 
they form a part, or not, of the required area. 



EXAMPLE. 



Let it be required to survey the tract ABCDFGH, which 
is bounded in part by the creek BCD and the river GFD. 



SURVEYING 



107 



Run the station lines 
BD and DG, closing with 
the rectilinear boundary 
BAHG. 

On BD measure the 
distances B&, BZ, Bm, Be> 
to perpendiculars, or off- 
sets, drawn from the line 
BD to the most noticeable 
flexures in the irregular 
side BOD. 

Also on the station line 
DG measure the distance 
Ds to a perpendicular drawn to a flexure at F 
DFG. 




in the side 



In computing the areas of the triangles and trapezoids con- 
tained between the station line BD and the side BCD, it will 
be most convenient to arrange the distances B&, BZ, &c, in 
tabular form ; thus, 



DISTANCES. 


OFFSETS. 


ALTITUDES. 


MULTIP, 


DOUBLE AEEAS. 


B&, 4.0 


1.5 


B&, 4.0 


1.5 


6.00 


BZ, 7.0 


1.0 


Tel, 3.0 


2.5 


7.50 


Bm, 10.5 


2.5 


Zm, 3.5 


3.5 


12.25 


Bo, 13.7 


2.0 


mo, 3.2 


4.5 


14.40 


BD, 17.5 


0.0 


oD, 3.8 


2.0 


7.60 



The second column contains the perpendiculars, or offsets, 
measured from the points k, Z, m, o. The next column con- 
tains the altitudes of the triangles and trapezoids ; the alti- 
tude hi being found by subtracting BJc from BZ; Irn by 
subtracting BZ from Bm, &c. 

The column of multipliers contains the oases of the triangles, 
and the sum of the two parallel sides of each trapezoid. 
These sums are obtained by adding together each two con- 
secutive offsets. 

The products of the corresponding altitudes and multipliers 
are the double areas of the respective triangles and trapezoids. 



108 BOOK III. 

Half the sum of these double areas must be added to the 
area contained in the rectilinear figure ABDGH. 

The area cut off by the other station line DC may be found 
by means of an offset at s. This area must also be added to 
that of the rectilinear figure. 



DIVISION" OF LAND ; 

OR, OF POLYGONS IN GENERAL. 

In the following Problems the sides and angles of the 
polygons are supposed to be given, though it will not be 
necessary, in every case, to use all these parts. 

From the hearings of the sides the interior angles of a 
survey may be readily determined (85) ; and, conversely, the 
bearings of the sides may be determined from the interior 
angles, provided the bearing of any one side is known (92). 

The computations required in these Problems are such as 
the student is already supposed to be familiar with ; it is 
therefore considered sufficient to show the methods of solu- 
tion, without any numerical operations. 

PKOBLEM I. 

(94.) From a given Triangle to cut off am/ given area, by 
a straight line drawn from its vertex to the base. 

Let ABC be the given triangle 
from which is required to cut off 
the triangle ADC containing a 
given Area. 

To find the length of AD. a~ if 

The area ABC : the area ADC : : AB : AD (Geom. 208). 

Having found AD, the other parts of the triangle ADC 
might also be computed (57) ; and the bearing of the dividing 
line CD may be readily determined when the bearing of any 
one of the sides of ABC is known. 




SURVEYING 



109 




PROBLEM H. 

(95.) From a given Triangle to cut off any given area, by 
a straight line drawn parallel to one of its sides. 

Let DFG be a given triangle 
from which it is required to cut 
off the triangle IKG, containing 
a given area, by the straight line 
IK parallel to DF. 

To find the length of IG. The Triangles DFG and IKG 
are similar (Geom. 222), and we therefore have 

The area DFG : the area IKG ; : DG 2 : IG 2 (Geom. 242). 

The square root of the numerial value of IG 2 will be the 
length of IG. 

The angle GIK is equal to the angle D ; and the remaining 
parts of the triangle GIK may now be computed (55). 




PROBLEM III. 

(96.) From a given Triangle to cut off any given area, by 
a straight line drawn from a given point in one of the sides. 

Let ABC be a given triangle 
from which it is required to cut 
off the triangle ADF, containing 
a given area, by the straight line 
DF drawn from the given point 
D in the side AB. 

The point D being given, it will be understood that the 
distance AD is given, and that the distance AF is to be 
determined. 

The area ABC : the area ADF : : AB . AC : AD . AF 
(Geom. 241). 

The numerical value of the product AD . AF divided by 
AD will give the length of AF. The other parts of the tri- 
angle ADF may also be computed (57). 




110 BOOK III. 

\ 

PROBLEM IV. 

(97.) The angle contained between two straight lines bei/ng 
given, to cut off any given area by a straight line drawn from 
a given point in one of those lines. 

Let BAC be a given angle, and 
B a given point in the side AB ; 
and let it be required to cnt off 
the triangle ABD containing a 
given area. 

The point B being given, it will be understood that the 
distance AB is given, and that AD is to be determined. 

Sin A . AB : 2 area ABD : : Kadius : AD. 

Having thus determined AD, all the remaining parts of 
the triangle ABD may be computed (57). 

The correctness of the preceding proportion may be shown 
thus : We have 

Kadius : sin A ; ; AB . AD : 2 area ABD (65) ; 

K . AB : sin A . AB ; : AB . AD : 2 ABD (Geom. 184). 

By interchanging each antecedent and its consequent 
(G-eom. 182), and then interchanging the two means in the 
resulting proportion (Geom. 183), we find 

sin A. AB : 2 ABD : : E . AB : AB . AD : K : AD 
(Geom. 185). 

PROBLEM V. 

(98.) The angle contained between two straight lines being 
given, to cut off any given area by a straight line which shall 
make a given angle with one of those lines. 

Let BAC be a given angle, and let 
it be required to cut off the triangle 
ADF, containing a given area, by 
the straight line DF making a giv- 
en angle ADF with the line AB. 




SURVEYING. HI 

The angle AFD will be found by subtracting the sum of 
the other two angles of the triangle ADF from 180°. Then, 
in this triangle, 

sin A . sin D : radius . sin F : : 2 area ADF ; AD 2 (66), 

The square-root of the numerical value of AD 2 will be the 
length of AD. The other parts of the triangle ADF may 
then be computed (55). 

PROBLEM VI. 

(99.) From a given Triangle to cut off any given area, 
towards a given angle, by the shortest line possible. 

Let ABC be a given triangle 
from which it is required to cut 
off the triangle ADF, containing 
a given area, by the shortest line 
DF that will cut off such area, 
towards the angle A. 

By one of the preceding Problems cut off any triangle Ars 
containing the given area. Thus we may assume any dis- 
tance Ar on the side AB, and then, by Problem IV. (97). 

Sin A . Ar : 2 area Ars '. '. Radius ! As. 

Take AD and AF each equal to a mean proportional 
between Ar and As (Geom. 179) ; draw the straight line DF, 
and it will be the shortest that can be drawn between the 
sides AB and AC, cutting off the given area in the triangle 
ADF. 

The demonstration is as follows : 

First. The triangle ADF is equivalent to Ars, and thus 
contains the given area, because AD 2 or AD . AF is equal to 
Ar . As (Geom. 178), and said area would be determined 
from either of these products, together with Radius and Sin 
A (65). 

Secondly. The line DF, cutting off the equal parts AD and 
AF, is shorter than rs, cutting off the unequal parts Ar and 
As, of AB and AC. 




112 



BOOK III. 




The line rs cannot pass through 
the middle point m of the line 
DF, for then the triangle ADF 
would be less than Ars (Geom. 
494), but to make the former tri- 
angle equivalent to the latter, the 

perpendicular Am (Geom. 50), must evidently cross the line 
rs. Having drawn Ax perpendicular to rs produced, we have 
Ax less than Ao (Geom. 57), and therefore less than Am ;_ 
and since the triangles Ars and ADF are equivalent, while 
the altitude of the former is less than that of the latter, it 
follows that the base rs of the former is longer than the base 
DF of the latter (Geom. 206). 



PROBLEM VII. 



(lOO.) From a given Triangle to cut off any given area hy 
a straight line drawn through a given point within the tri- 
angle. 

Let ABC be a given triangle 

from which it is required to cut 

off the triangle ADF, containing 

a given area, by the straight 

line DF drawn through the point 

p -rc jn, — -Q- 

The point P being given in position, the perpendiculars 
Pm and ~Po will be given, or may readily be computed by 
Trigonometry. 

Let a represent the given area ADF ; then the length of 
AD may be computed from the formula, 




AD = — ± t /( 

Pm V \Pm 2 



2 To. a 



Pm . nat. sin DAF> 



This formula is obtained in the following manner. By 
making the Radius unity, we shall find the area ADF to be 



a=i AD . AF . nat. sin DAF (65). 



SURVEYING. 113 

But the area ADF is also equal to the sum of the areas 
ADP and APF ; that is, 

a=i AD . Pm+1 AF . To (Geom. 206). 

The first of these two equations gives 

2a 



AF 



AD . nat. sin DAF 



By substituting this value of AF in the second equation, 
we obtain 

t atn -n , a. To 

a=i AD . Pra+ 



AD . nat. sin DAF 



Multiplying both sides of this equation by the denominator 
in the second member, we shall obtain an equation of the 
second degree with respect to AD, from which AD will 
be found, as in the formula already given (Algebra 260). 



problem vm. 



(101.) From a given Parallelogram or Trapezoid to cut 
off a parallelogram containing a given area. 



Let ABCD be a given trapezoid, d, ^~ 

from which it is required to cut off the 
parallelogram AG, containing a given 
area, by the straight line FG parallel 



to AD. P 

The given area of AG divided by AD, regarded as the lose, 
will give the perpendicular IF, or altitude of the parallelo- 
gram AG (Geom. 205). Then in the right-angled triangle 
AIF, we shall have the angle A and the perpendicular IF to 
find AF, equal to DG. 

Eadius : IF : ; sec AFI : AF (46) ; 
or sin A : IF ; ; sin 90° : AF (48) ; 

The same solution would evidently be applicable if AC 
were a parallelogram. 



114 BOOK III. 



PROBLEM IX. 




(102.) From a given Parallelogram or Trapezoid to cut 
off a trapezoid containing a given area, by a straight line 
making a given angle with one of the parallel sides. 

Let ABCD be a given trapezoid, 
from which it is required to cut off a 
given area AG by a straight line FG, 
making a given angle AFG with the 
side AB, which is parallel to DC. 

Subtract the sum of the two angles DAB and AEG from 
180° ; and make the angle ADK equal to what remains ; then 
DK will be parallel 'to FG, since the angle AKD will thus 
be made equal to AFG (Geom. 38 and 31). 

Compute the area of the triangle ADK, from the side AD 
and the angles (66). Subtract this area from the given area 
of AG, and the remainder will be the area of the parallelo- 
gram KG. 

In the triangle ADK, find the side DK (55). Divide the 
area of KG by the length of the base DK ; the quotient will 
be the altitude IG. 

The angle ADK subtracted from ADC leaves the angle 
IDG. Then in the right-angled triangle DIG we shall have 
the angles and the perpendicular IG, to find DG, equal to 
KF. 

Sin IDG : IG : : sin 90° : DG (55). 

The line FG may be drawn by the angle DGF, which is 
180°-AFG (Geom. 27). 

(103.) Scholium. — If, in the preceeding Problem, the 
given angle AFG were a right angle, the dividing line FG 
would be the shortest by which the given area AFGD would 
be cut off from the given trapezoid ABCD. 



SURVEYING. 115 



PROBLEM X. 

(104L.) .From a given Parallelogram or Trapezoid to cut 
off any given area by a straight line drawn between the 
parallel sides, and through a given point within the figure. 



Let ABCD be a given trapezoid, 
from which it is required to cut off 
a given area AFGD, by a straight 
line FG drawn between the two 
parallel sides AB and DC, and A ^i""3 

through a given point P within the figure. 



We may suppose the point P to be given by the angle 
AP and the distance AP. 
FAD leaves the angle PAD. 



FAP and the distance AP. This angle subtracted from 



Find tfce area of the triangle APD (65), the angle ADP, 
and the side DP (57). The angle ADP subtracted from 
ADC leaves the angle PDC. 



i t>- 



Through the point P draw the straight line mo perpen- 
dicular to AB and DC (G. 28.) ; and in the right-angled 
triangles APm and DP<? find the perpendiculars Pm and Yo 
(54). 

Let a represent the given area AFGD, and b the area of 
the triangle APD ; then the length of AF may be computed 
from the formula, 

. _, 25 . mo+2a (Po—mo) 

AF = — ; . 

(Fo— Pm) mo 

This formula is obtained in the following manner. ¥e 
have 

a=\mo (AF+DG)=4 mo . AF+i mo . DG (Geom. 207). 

But AFGD is equal to the sum of the triangles AFP, 
APD, DPG ; hence 

a=i Pm . AF+6+i Po . DG (Geom. 206). 



116 BOOK III. 

From the first of these two equations, 

mo 

By substituting this value in the second equation, and 
multiplying both sides by 2 mo, we shall obtain 

2# . mo=Pm . mo . AF+25 . mo+2a . ~Po— ~Po .mo.AF; 

from which the value of AF will be found. 



PROBLEM XI. 

(105.) From any given Quadrilateral to cut off a quadri- 
lateral containing a given area, oy a straight line making a 
given angle with one of its sides. 

Let ABCD be a given 
quadrilateral, from which it 9r*\ 

is required to cut off the l)x^\ \ 

quadrilateral AG, containing ^ \ \ \ 

a given area, by the straight .„-•*' I \ \ 

line FG, making a given an- if i :f b 

gle AFG with the side AB. 

It is to be understood that the sides AB and CD are not 
parallel ; if they were, the Problem would be the same as 
Prob. IX. 

Produce the sides BA and CD until they meet in the point 
K, thus forming the exterior triangle ADK. The angle BAD 
subtracted from 180° leaves the angle KAD, and ADC sub- 
tracted from 180° leaves ADK. 

Compute the area of the triangle ADK from the side AD 
and the angles (66) ; add this area to the given area AG ; 
the sum will be the area of the triangle KFG. In the tri- 
angle ADK the angle K will be known, and the area KFG 
may be cut off by Problem V. "We shall thus find KF. In 
the triangle ADK find the side AK ; subtract AK from KF, 
and the remainder will be AF. 



SURVEYING. 



117 




PROBLEM Xn. 

(106.) From any given Quadrilateral to cut off a quadri- 
lateral containing a given area by the shortest line possible. 

Let ABCD be a given 
quadrilateral, from which it 
is required to cut off the 
quadrilateral AG, containing 
a given area, by the shortest 
line FG that will cut off such 
area, towards AD. 

It is to be understood that the sides AB and CD are not 
parallel ; if they were, the Problem would be solved in the 
same manner as Prob. IX. — the line FG only requiring to be 
drawn at right angles with the parallels AB and CD. 

Produce the sides BA and CD until they meet in K, thus 
forming the exterior triangle ADK. As in the preceding 
problem, compute the area ADK and add it to the given area 
AG ; we shall then have the area KFG ; and the side KF, 
equal to KG, may be found by Problem YI. AK, computed 
in the triangle ADK, and subtracted from KF, leaves AF. 



PROBLEM XIII. 

(l©7.) From any given Quadrilateral to cut off a quadri- 
lateral containing a given area, by a straight line drawn 
through a given point within the figure. 

Let ABCD be a given 
quadrilateral, from which it 
is required to cut off the 
quadrilateral AG, contain- 
ing a given area, by the 
straight line FG, drawn 
through the point P within the figure. 

It is to be understood that the sides AB and CD are not 
parallel ; if they were, the problem would be the same as 
Prob. X. 



£>•---- 




118 BOOK III. 

Produce the sides BA and CD until they meet in K, thus 
forming the exterior triangle ADK. As in the two preceding 
problems, compute the area ADK, and add it to the given 
area AG ; we shall then have the area KFG ; and the side 
KF may be found by Problem TIL Then AK, computed 
in the triangle ADK, and subtracted from KF, will leave 
AF. 



PEOBLEM XIV. 

(lO§.) From any given Polygon to cut off a given area, hy 



a 



straight line drawn from a given point in one of its sides. 




Let ABCDFG be a given polygon, 
from which it is required to cut off 
a given area AI, by a straight line 
PI, drawn from a given point P in 
the side AB. 

By comparing the area to be cut 
off with that of the whole figure, we 
may generally determine, by mere 

inspection, on which side of the polygon the point I will fall. 
Suppose that side to be FD. 

Draw the straight line PF, cutting off an area less than 
AI. We may suppose the bearings of the sides AB, AG, 
GF, FD to be known ; or, from the interior angles, we may 
determine the bearings of these sides with reference to an 
assumed meridian (91). The distance AP being given, we 
shall have, in the polygon AF, the bearings, real or assumed, 
and the lengths of all the sides, except the bearing and length 
ofFP. 

Find the bearing and length of FP (88), and the area AF 
(89). Subtract this area from the given area AI ; the 
remainder will be the area of the triangle PFI. 

From the bearings of FP and FD find the angle PFD (85). 
Then having the given point P in the line PF, we may find, 
by Problem IY., the length of FI, so that the triangle PFI 
shall contain the area already found for it, and AI the area 
required to be cut off from the given polygon. In the triangle 



SURVEYING. 119 

PFI we may also compute the side PI and the remaining 
angles (57). 

PROBLEM XV. 

(109.) From any given Polygon to cut off a given area by 
a straight line making a given angle with one of the sides. 

Let ABCDFGH be a given poly- t?^^-"^? 

gon from which it is required to cut 
off a given area AI, by a straight G f 
line KI, making a given angle AKI U 
with the side AB. \ 

By comparing the area to be cut 
off with that of the whole figure, we 

may generally determine, by mere inspection, on which side 
of the polygon the point I will fall. Suppose that side to be 

m 

Draw the straight line AF, cutting off the figure AFGH 
less than the given area AKIFG1L "We may suppose the 
bearings of the sides AH, HG, GF to be known ; or, from 
the interior angles, we may determine the bearings of these 
sides with reference to an assumed meridian (91). In the 
polygon AHGF we shall then have the bearings, real or 
assumed, and the lengths of all the sides, except the bearing 
and length of FA. 

Find the bearing and length of FA (88), and the area 
AFGH. Subtract this area from the given area AKIFGH ; 
the remainder will be the area of the quadrilateral AKIF. 
This quadrilateral, with its area thus determined, may then 
be cut off by the method of Problem XL ; and thus the given 
area AKIFGH will be cut off from the given polygon. 

(110.) Scholium. — If the line KI were required to be the 
shortest that could be drawn from AB to FD, cutting off the 
given area ; the quadrilateral AI might be cut off by Pro- 
blem XII; and if the line KI were required to be drawn 
through a given point within the polygon ABC, &c, the 
quadrilateral AI might be cut off by Problem XIII. 



120 - BOOK III. 



Additional Instructions for the Division of Polygons. 

(ill.) When a given polygon is required to be divided 
into two parts having a given ratio to each other, — find the 
areas of the parts, and then divide the polygon according to 
these areas. 

For example, to divide a polygon containing a given area, 
A, into two parts which shall be to each other in the ratio of 
b to c. 

_ , , .. A . A. b z , ... A.c 

o+c : b A : — ; b+c : c : : a : — ; 

b+c b+c 

(Arithmetic, 147 and 148). 

The areas of the two parts will therefore be A. b-7-(b+c) and 
A.c-7-(b+c); and the given polygon will be divided in the 
given ratio, by cutting off from it either of these areas, as 
in any of the preceding Problems. 

2. When a given polygon is required to be divided into 
three or more parts, cut off from it one of the parts, and then 
from the remaining polygon cut off another part, and so on. 

To divide a polygon containing a given area, A, into three 
parts which shall be to one another as the numbers 5, c, 
and d. 

The areas of the three parts will be found to be 

A.b A.c A.d 



~> Z T' 



b+c+d b+c+d b+c+d 

The method of dividing the given polygon will therefore 
be, to cut off' the first of these areas, and then from the 
remaining polygon to cut off the second area ; the polygon 
still remaining will contain the third area. 

By a proper application of the methods developed in the 
preceding Problems, the surveyor will be able to execute 
most of the cases that will be likely to occur in the Division 
of Land. 



SURVEYING. 121 



VARIATION OF THE MAGNETIC NEEDLE. 

(112.) The Variation, or declination, of the magnetic 
needle-^ at any place, is the angle which the magyietic me- 
ridian of that place (76) makes with the true meridian 
passing through that place and the poles of the Earth. This 
variation is said to be east, or west, according as the north 
end of the needle points east, or west, of the true meridian. 

The variation of the needle is not the same at all places ; 
nor does it continue the same at any one place for any con- 
siderable length of time. 

It is often necessary in Surveying that the amount of 
magnetic variation should be known; we shall therefore 
explain a method by which it may be determined with the 
Compass. 

The Polar Star. 

(113.) A well known star, called the Worth Star, is about 
li° from the true novth. pole of the heavens, or point in which 
the Earth's axis produced intersects the starry concave ; and 
apparently revolves around the pole in 23 A. 56m. 

When directly above, or below, the pole, this star is in the 
plane of the true meridian y and the Compass directed to a 
vertical line in range with the star, in either of these two 
positions, would show the variation of the needle. But the 
star immediately departs from the plane of the meridian ; and 
the computed time of its meridian position would not ordi- 
narily be given with sufficient accuracy by the clocks or 
watches in common use. 

When at its greatest eastern or western elongation from the 
pole, the star's motion for several minutes is nearly vertical ; 
and a deliberate observation may then be made upon it, for 
determining the variation. 

The times for making these observations are given, for 
every 10th day in the year, in the following Tables of the 
North Star's greatest elongations. 



122 



BOOK III. 



Eastern Elongations. 



DAYS. 


APRIL. 


MAY. 


JUNE. 


JTJLY. 


AUG. 


BEPT. 




H. M. 


H. M, 


H. M. 


H.M. 


n. m. 


H. M. 


1st, 


6 32 A.M. 


4 34 A.M. 


2 33 A.M. 


35 A.M. 


10 30 P.M. 


8 28 P.M. 


11th, 


5 53 A.M. 


3 55 A.M. 


1 53 A.M. 


11 52 P.M. 


9 51 P.M. 


7 49 P.M. 


21st, 


5 14 A.M. 


3 16 A.M. 


1 14 A.M. 


11 13 P.M. 


9 11 P.M. 


7 09 P.M. 



Western Elongations. 



DAYS. 


OCT. 


NOT. 


DEC. 


JAN. 


FEB. 


MAKCH. 




11. M. 


H. M. 


11. M. 


H. M. 


H. M. 


H.M. 


1st, 


6 22 A.M. 


4 21 A.M. 


2 22 A.M. 


19 AM. 


10 13 P.M. 


8 22 P.M. 


11th, 


5 43 A.M. 


3 41 A.M. 


1 43 A.M. 


11 35 P.M. 


9 33 P.M. 


7 43 P.M. 


21st, 


5 04 A.M. 


3 02 A.M. 


1 0-1 A.M. 


10 56 P.M. 


8 54- P.M. 


7 04 P.M 



The preceding Tables have been taken from one first 
published by Professor W. M. Gillespie, in h*is valuable 
" Treatise on Land-Surveying." They are strictly correct 
only for Latitude 40°, and the date July 1st, 1854 ; yet they 
will be found to answer the purpose for which they are used 
in determining the magnetic variation, in any part of the 
United States, for many years to come. 

It will be observed that only those elongations have been 
given in our Tables which occur in the night, when the polar 
star is visible. 

The purpose for which an observation is made on the polar 
star, at its greatest eastern or western elongation, is, to deter- 
mine its magnetic bearing in that position, and thence to 
deduce the variation of the needle. 

The Practical Procedure. 



(114.) 1. -Nail a plank, about 3 feet long and 6 or 8 inches 
wide, and planed smooth on its upper side, in a horizontal 



SURVEYING. 123 

position, east and west, on two firm supports, elevating it 
three or four feet from the ground. 

2. At the distance of eight or ten feet, towards the north, 
plant a stiff pule or stake, having a cross piece at its top, 
from which a plumb-line is suspended. The stake must be 
of such height that its top shall appear a few inches above 
the star, viewed from the plank. 

3. On the plank set a staff or standard, two or three feet 
high, fastened firmly to a block, planed smooth at its base, 
and having a cross-piece at its top, from which a plumb-line 
is suspended. 

4. About twenty minutes before the time of elongation, in 
one of the preceding Tables, move the standard on the plank 
until the polar star ranges with the two plumb-lines. Keep 
the star in range with the two lines by still moving the 
standard, until the star, for several minutes, is found to 
remain behind the lines, without any easterly or westerly 
motion. The star will then be at its greatest elongation, and 
its magnetic bearing, in that position, will be shown by the 
Compass, with its sights directed in a range with the two 
plumb-lines. 

As this observation is to be made at night, the plumb lines 
must be rendered visible by lights held near them. Steadi- 
ness of the plumbs will be secured by causing them to hang 
in vessels filled with water. The observation with the Com- 
pass may be deferred until day-light. 

The magnetic bearing of the !N"orth Star, found as above, 
will be the magnetic bearing of a horizontal line lying in the 
vertical plane which passes through the star, at its greatest 
eastern or western elongation, and the observer's position on 
the Earth. This bearing, compared with the true bearing of 
the same line, will show the magnetic variation. The true 
bearing must be computed from the latitude of the place and 
the polar distance of the star. This polar distance will be 
found by subtracting the star's declination (given in the 
American Almanac) from 90°. 



124: BOOK III 



EXAMPLE. 



At a place in latitude 38° N. the magnetic bearing of the 
North Star, at its greatest western elongation, on the 1st day 
of March, 1857, was found by observation to be JST. 3° 15' E. 
"What was the variation of the magnetic needle f 

To compute the true hearing of the star at the time of the 
observation. — By referring to the American Almanac for 
185T, page 55, we find that the declination of the North Star 
{Polaris), on the first day of March, was 

88° 33' 6.4". 

Then its polar distance was 90° - (88° 33' 6.4") = 1° 26' 
53.6". 

Cosine of latitude 38° 9.896532 

is to Kadius 10 10 10.000000 

as sine of polar distance 1° 26' 53.6" 8.402663 

is to the true bearing 1° 50' 16". . . 8.506131 

The star being at its greatest western elongation at the time 
of observation, 

Its true bearing was therefore 1° 50' 16" W. ; 
its magnetic bearing was 3° 15' 00" E. 

What then was the variation f The north end of the 
needle was turned, by the amount of the magnetic bearing, 
west of the star ; and the true meridian ran, by the amount 
of the true bearing, east of the star. The north end of the 
needle was thus turned west of the true meridian, by the sum 
of the two bearings ; that is, 

The variation was 5° V 16" W. 

The following will be a convenient Genercd Rule for finding 
theYariation from the two bearings : — Eegard an east bearing 
as positive, and a west bearing as negative / subtract the 
magnetic from the true bearing, algebraically ; the result 
will be the variation, East if positive, West if negative. 



SURVEYING. 125 

Proceeding in this manner with the two bearings in the 
Example, the result will be — (5° 5' 16"), the variation west. 

The Proportion for obtaining the true hearing involves 
principles in Spherical Trigonometry, which the student is 
here presumed not to have studied. A like proportion may 
always be employed in finding the variation of the magnetic 
needle, when the latitude of. the place is known, either 
accurately or approximatively. 

The observation for obtaining the magnetic bearing of the 
Polar Star, at its greatest eastern or western elongation, may 
be most easily and accurately made with an instrument, such 
as the Theodolite, in which a sight is taken through a Teles- 
cope that may be turned, either horizontally or vertically, 
upon the object. The method will readily occur to any one 
using such an instrument, and need not here be described 



o 



Magnetic Variation in the United States. 

(115.) The line of no variation is a line passing through 
those places at which the magnetic coincides with the true 
meridian. This line — according to the observations and 
researches of Professor Elias Loomis — in the year 1840, ran, 
nearly in a straight direction, from a little west of Cape 
Hatteras, 1ST. C, through the middle of Virginia, and the 
middle of Lake Erie. 

At all places east of this line the variation was west ; at 
all places west of this line the variation was east ; and the 
variation increased nearly in proportion to the distance from 
the line of no variation, amounting to 18° in Maine, and 21° 
in Oregon. 

" Since 1840, the variation in New England has increased 
about five minutes annually; in New York and Pennsyl- 
vania it has increased from three to four minutes annually. 
In the Western States it decreases at about the same rate, 
and in the Southern States it decreases about two minutes 
annually*" 



126 BOOK III. 

Diurnal and Irregular Variations. 

(116.) The North end of the magnetic needle is found to 
move westward, in the Northern Hemisphere, from about 
8 o'clock, A. M., until 2 P. M., and then gradually to return 
to its former position. The amount of this diurnal variation 
is from 10' to 15' in Summer, and about half as much in 
Winter. A similar but smaller change occurs during the 
night. In the Southern Hemisphere the diurnal motion is 
in the opposite direction. 

The direction of the magnetic needle is also subject to 
considerable disturbance during the time of a thunder storm, 
or of an Aurora Borealis ; and sudden but transient changes 
of direction have sometimes been observed in it, unconnected 
with any known law whatever. 



Application of the Magnetic Variation. 

(117.) The variation of the magnetic needle should be 
recorded in connection with every survey made with the 
Compass. The sides of the survey may then be retraced, 
from the recorded bearings, at any subsequent time, by 
making the proper allowance for the change of variation. 

As this record of the magnetic variation is often omitted, 
it is necessary to be able to determine the change of variation 
between the times of the two surveys, independently of the 
absolute variation. This may be done by means of the 
recorded bearing from any one known point or corner of the 
survey to another. 

EXAMPLE. 

In the Field-Notes of an old survey I find that the magnetic 
bearing from one known corner to another was N. 10° E. 
What change of variation in the magnetic needle has occurred 
since the field-notes of that survey were taken ? 



SURVEYING 



127 



Let A and B be the two corners, the recorded 
bearing of the line AB being K 10° E. Set the 
Compass at A, with its sights adjusted to the same 
bearing, as indicated by the needle, and suppose 
the sights, from the change of variation, to point 
in the direction AC. 

Measure AC to the point at which it meets BC 
perpendicular to either AC or AB ; also measure 
BC. Then in the right-angled triangle ABC two 
sides, AC and BC, will be known — from which the 
angle BAC may be computed. This angle will 
be the change of variation. 

Tlie direction, east or west, in which the change has 
occurred, will be known from the line AC's running east, or 
west, of AB. 

In this Example it is implied that the corner B cannot be 
seen from A, — otherwise the change of variation would 
become known by merely adjusting the Compass sights to 
the line AB, and noticing the present bearing. 



To retrace the line AB. — Suppose the change of variation, 
BAC, to be 15° west / the Compass sights will be adjusted 
to AB when the bearing indicated is N. 10° +15° E., that is, 
~R. 25° E. — By sighting, at each corner, 15° more towards 
east than the recorded bearing expresses, all the sides of the 
survey may be retraced. 



The inconvenience of having to make this allowance in 
directing the sights for each side of the survey, is obviated 
by the Vernier Compass (78), in which the graduated circle, 
by a movement around its centre, may be permanently 
adjusted to the change of variation. Its indications will 
then be the same as if no change had occurred; and the 
sides will all be retraced by their recorded bearings. 



128 BOOK III. 

SURVEYS OF THE PUBLIC LANDS. 

(ll§.) The Public Lands of the United States, when sur- 
veyed, are divided, by true meridian lines and parallels of 
latitude, as nearly as may be, into Townships six miles square, 
each township into Sections one mile square, each section 
into Quarter-sections, and sometimes into Eighths and Six- 
teenths of a Section. 

The Surveyor begins with establishing, astronomically, a 
principal meridian and an east and west "base line inter- 
secting it within the territory ; also standard parallels of 
latitude, north and south of the base line, at intervals 
embracing the length of 4 or 5 townships. 

By means of the Compass, with allowance for magnetic 
variation, other meridians are run from the "base line, north 
and south, at intervals of six miles on that line. The 
meridians converge slightly in going northward, and diverge 
slightly in going southward, since these surveys are in the 
Northern Hemisphere. On arriving at each standard parallel, 
the Surveyor therefore starts the meridians anew, at inter- 
vals of six miles on the parallel ; and thus a near equality of 
surface is preserved between equal lengths of the meridians. 

While thus proceeding northward, or southward, with a 
meridian, at intervals of six miles on it, parallels of latitude 
are run, with the Compass, between that meridian and the 
preceding one, and thus the territory is divided into Town- 
ships, by north and south, and east and west lines. 

A Township, it is thus seen, will be slightly trapezoidal in 
form, and will contain about 6x6=36 square miles, or 23040 
acres, — a little under, or a little over, this quantity, according 
as it lies north, or south, of the base line. 

Each Township is divided, by north and south, and east 
and west lines, drawn at equal intervals between the town- 
ship boundaries, into 36 sections, containing each about one 
square mile, or 640 acres. The subdivisions of the Sections 
into Quarter-sections, Eighths, and Sixteenths are also made 
by nortli and south, and east and west lines. 



SURVEYING. 



129 



Method of Designating Townships and Sections. 

(119.) A range of Townships consists of those which lie 
north or south of one another ; and the successive ranges are 
designated by the numbers 1, 2, 3, &c, beginning at the 
principal meridian, and proceeding eastward, and westward. 
The successive Townships in the same range are designated 
by the numbers 1, 2, 3, &c., beginning at the hose line, and 
proceeding northward, and southward. 

Thus Township 2 north, range 3 east, designates the 2d 
township north of the base line, in the 3d range of town- 
ships east of the principal meridian. 



w 



E 



The 36 sections in each Town- jf 

ship are numbered 1, 2, 3, &c, 
beginning at the north-east sec- 
tion, and going west from 1 to 6, 
then east from 7 to 12, then 
west from 13 to 18, and so on, 
to 36, which is the south-east 
section. 

The quarter-sections are de- 
signated by the terms north- 
east, north-west, south-east, and 

south-west. Thus, the particular locality of a quarter-section 
would be designated by saying, the south-west quarter of 
section 15, township 3 south, range 4 west. 



6 


5 


4 


3 


2 


1 


1 


8 


9 


10 


11 


12 


18 


17 


16 


15 


14 


13 


19 


20 


21 


22 


23 


24 


30 


29 


28 


21 


26 


25 


31 


32 


33 


34 


35 


36 



It is the business of the Surveyor to mark the corners of 
townships, sections, and quarter-sections, by blazing a tree, 
planting a post or stone, or by raising a mound of earth where 
the corner has been established. These corners furnish the 
means of precisely identifying any township or section. 

The preceding method of laying out the Public Lands was 
introduced in the year 1802, by Colonel Mansfield, then 
Surveyor-General of the North- Western Territory. It secures 
the titles of individual purchasers of such lands against all 
controversy in regard to their location. 



130 B K 1 1 1 



LEYELING- 



(120.) Leveling is the method of determining the distance 
that any given point is above or below a level surface con- 
ceived to pass through any other given point of the surface 
of the earth, — this distance being called the difference of 
level of the two points. 

A level surface, as before observed (71), is nearly spherical; 
for the distances embraced in leveling it is considered per- 
fectly so, and the difference of level between any two points 
thus becomes equal to the difference of their distances from 
the centre of the earth. 

A line of true level is any line which lies entirely in a 
level surface ; a line of apparent level is the same as a 
horizontal line passing through the point of observation 
(61... 2). Points are on the same level when they are in the 
same level surface. 

• Leveling is required in establishing the lines of roads and 
canals, in determining the points to which the water of a 
distant fountain or stream may be conducted in pipes or 
aqueducts, and for other useful purposes, 

LEVELING INSTEUMENT. 



(121.) The Leveling Instrument, or, simply, the Level, is 
used to determine a line of apparent level. It consists of a 
Telescope, T S, with a spirit-level attached to it to show its 
horizontal adjustment, and various other appendages to 
facilitate its accurate use. 

It is supported by a tripod under the plate AB. Standing 
on this plate are four screws, which pass through the plate cd, 



SURVEYING. 131 

and which, on being turned, two and two, as occasion re- 
quires, change the inclination of the plate cd, and thus serve 
to adjust the instrument. 

The axis of the telescope being parallel to the spirit-level, 
the instrument will be adjusted to a horizontal plane when, 
on turning the bar EF around the vertical axis of the instru- 
ment, the bubble of air in the tube of the spirit-level is found 
to remain in the centre. 

The line of collimation in the telescope, the same with its 
axis, is a straight line passing through the centres of its eye 
and object glasses. The direction of this line from the centre 
of the eye-glass is shown by the intersection of a horizontal 
and a vertical spider's line, or wire, in the common focus of 
the two lenses. The intersection of the cross-wires marks, on 
any object viewed, the precise point to which the axis of the 
telescope is directed. 

Leveling Bods. 

(122.) A Leveling Rod is an instrument specially adapted 
for measuring the height from any assumed point of the 
earth's surface to a line of apparent level sighted through a 
telescope. 

It usually consist of two rectangular staves in contact, and 
capable of sliding on each other by means of a groove in one 
of them, with a movable target, which may be brought into 
a line with the axis of the telescope when the rod is held 
vertically on the ground, at a distance. 

The target (represented, with a part of the 
rod, by the accompanying figure) is divided 
into four equal spaces by a horizontal and a 
vertical line. These spaces are painted, alter- 
nately, white and red, or white and black, in 
order to a more distinct visibility at a distance. 

When drawn out to its full extent, the rod 
is usually twelve feet in length, and is divided 
to hundredths of a foot. The target has an 
opening in its face through which the divi* 




132 BOOK III. 

sions of the rod are seen, and on one 
indicates thousandths of a foot on the rod. 



sions of the rod are seen, and on one side of which a vernier 



Practical Leveling. 

(123.) Two leveling rods are held vertically on the ground 
at the points whose difference of level is required. The 
telescope of the Level, adjusted to a horizontal plane, is 
turned first to one and then to the other rod, and the height 
of the visual line through the intersection of the cross wires 
is measured on each rod, by means of the target and vernier. 

The difference between these heights is the difference of 
level between the two points, or stations, provided they be 
equidistant from the telescope. When not thus equidistant, 
the observed heights are reduced, by computation, to a true 
level with the telescope ; the difference of the results is the 
difference of level of the two stations. 




Let am and ho be two leveling rods held vertically at the 
points a and b on the earth, and let the leveling telescope be 
set midway between them. The visual line mo is a line of 
apparent level, but it is plain that the points m and o are 
equidistant from the centre of the earth ; so that am— bo is 
the difference of the distances of the points a and b from the 
centre, and therefore the difference of level of these points. 

But the points m and r are plainly not equidistant from 
the centre of the earth, and hence the difference of the heights 
am and or is not the difference of the distances of a and o 
from the centre. 

Let st be the true level of the telescope, and let am and cr 
be reduced, by computation, to as and ct (72); then the 
difference between as and ct is evidently the difference of the 
distances of a and c from the centre of the earth, or the 
difference of level of those points. 



SURVEYING. 133 

The quantity to be subtracted from each of the observed 
heights, am, cr, to find the true level of the telescope may 
be computed thus : The true level st deviates from the 
apparent level mr proportionably to the square of the distance 
from the telescope, and amounts to 8 inches at the distance 
of 1 mile (72). Suppose that we want this deviation for the 
distance of 20 chains. One mile being 80 chains, we have 

80 2 : 20 2 ; : 8 inches \ 0.5 inch. ; 

the observed height must therefore be diminished half an 
inch at the distance of 20 chains. In like manner the proper 
reduction may be computed for any other distance. 

(124.) When the two stations whose difference of level is 
required are invisible from each other, level heights are 
measured, as above described (123), for the first station and 
an intermediate station ; then for this second station and 
another intermediate one, and so on to the last station. 

All the heights measured on the side of the last station, 
from the Leveling Instrument — called the forward sights, — 
are added into one sum, and those on the side of the first 
station, called the bach sights, into another ; the difference 
between these two sums is the difference of level between 
the two extreme stations. 



TOPOGEAPHT. 

(125.) Topogeapht is a description and delineation of any 
place, or small portion of the earth's surface, including its 
inequalities of hill and vale, and whatever noticeable objects 
that surface presents. 

The inequalities or undulations of the ground may be 
indicated by a system of curved lines which represent the 
intersections of the surface by a series of equidistant horizontal 
planes ; these curves falling nearer together in proportion to 



134: 



BOOK III. 



the greater steepness of the slopes to be delineated. Such a 
representation forms a 



TOPOGRAPHICAL MAP. 




Let the curves marked 0, 5, 10, &c, be the intersections 
of the surface of an uneven piece of ground by a series of 
horizontal planes, at equal intervals, say five feet, one above 
another ; these curves show that the surface ascends within 
the lines marked 0, having summits at a and g. 

In the projection of these curves on the same plane, it is 
evident that they will be nearest to each other where the 
slopes are steepest; so that the relative distances of these lines 
will indicate the varying acclivities of the surface. In shad- 
ing the map, the steepest slopes are made darkest, since these 
would be least illuminated by the sun shining upon the sur- 
face from above. 



We shall briefly indicate but one of the methods which 
may be pursued in making the necessary observations and 
measurements on the ground. 

From the summit a a series of radiating lines, ab, ac, ad, 
may be run with the compass, and staked off on the ground. 
By means of the leveling instrument the level curves 0, 5, 
10, &c, may be run, and the intervals may be measured at 
which they intersect the lines ah, ae, ad. These points of 
intersection are to be marked, according to an assumed scale 
of distances, on the lines ab, ac, ad, in the map, and the 
curved lines drawn through them. 



SURVEYING. 



135 



GEODESIC SUKVEYESTG. 

(126.) Geodesic Surveying consists in those extensive 
observations and measurements on the earth which are 
necessary for delineating, or computing, any large portion of 
its surface, or for determining the figure and magnitude of 
the entire body of the earth itself. 

This is also called Trigonometrical Surveying, since its 
fundamental operations have reference to the measurement 
of a connected series of triangles. 

It is of these fundamental operations, chiefly, that a brief 
account will here be given. The higher procedures of this 
kind of surveying depend on principles of both mathematics 
and physics with which the student is here supposed to be 
unacquainted, and do not come within the province cf general 
education. 



Measurement of Angles. 

(127.) In Trigonometrical Surveys the numerous horizontal 
angles employed are to be measured with the greatest 
possible accuracy. For this purpose, the Theodolite is com- 
monly used. 




The Theodolite is furnished with a graduated circle AB, 
called the graduated limb, and a vernier circle cd } which 

7 



136 BOOK III. 

moves freely upon AB, while both these circles admit of a 
horizontal motion abont the vertical axis E. A vertical 
graduated circle, or arc, EG is firmly supported on the vernier 
plate cd. A telescope TS attached to the circle or arc EG, 
has a motion in common with this circle, about the centre of 
the latter, in a vertical plane, and also a horizontal motion in 
common with the vernier circle cd. 

To the axis E prolonged a tripod to support the instrument 
is attached, and also suitable contrivances, as in the Level 
(121), for adjusting the circles AB and cd to a horizontal posi- 
tion — the spirit level, under the telescope, showing when that 
position is attained. 

When the instrument is thus supported and adjusted, the 
telescope TS may be readily directed to any object that can 
be seen through it. Two small wires, intersecting each other 
in its focus, mark on the object the precise point to which 
the axis of the telescope is directed. 

When the telescope is turned horizontally from one object 
to another, the graduated limb AB remaining fixed, the hori- 
zontal angle subtended by the two objects is measured by the 
number of degrees on AB through which the vernier circle cd 
is thus moved. A vertical angle is measured by the number 
of degrees through which the vertical circle EG is moved by 
turning the axis of the telescope, vertically, from one side of 
that angle to the other. 

With the additioA of a compass circle above cd, and con- 
centric therewith, the Theodolite also becomes fin instru- 
ment for measuring the hearings of lines, or the angles they 

make with the magnetic meridians. 

r 

The graduated limb, and its accompanying vernier circle, 
in the Theodolite, are made larger or smaller in diameter, 
according to the degree of accuracy desired in the measure- 
ment of angles. In the Survey, now in progress, of the 
Atlantic Coast of the United States, the largest Theodolite 
used has a limb of 30 inches diameter. It is graduated into 
12ths of a degree, and its vernier indicates single seconds. 



URVEYING. 



137 



Triangulation of the Survey. 

(128.) A Trigonometrical Survey begins with the selection 
of a suitable locality for a lose line AB. This line having 
been measured, horizontally, with the utmost accuracy, a 
staff or signal is erected at each end of it, which shall be 
visible one from the other. Thus are marked the first two 
stations, A and B. 

A third station C is next 
chosen at which a staff or 
signal will be visible from 
each of the former stations ; 
the former signals being also 
visible from that station. 
The angle subtended, at each 
station, by the signals at the 
other two, is then measured 
with the Theodolite, and 
thus in the triangle ABC 
one side, and all the angles 
will be known. 

The other two sides in this 
triangle may be computed 
trigonometrically. One of these sides, BC, is taken for the 
base of a second triangle, a fourth station, D, having been 
selected for its vertex. All the angles of the triangle BCD 
are also measured. 

The sides CD and BD may be computed trigonometrically. 
One of these sides, BD, is taken for the base of a third 
triangle, the station E having been selected for its vertex ; 
and all the angles of the triangle BDE are measured. In 
like manner, the triangulation is extended over the entire 
surface to be surveyed. 




The base line AB is taken longer or shorter according to 
the extent of country to be embraced by the triangulation. 
In very large surveys it is desirable that it be as long as 
possible — several miles in extent. The stations are chosen, 



138 BOOK III. 



when practicable, at points whose geographical position it is 
desirable to determine, such as mountain peaks, capes, or 
promontories, heads of bays, islands, &c. 



Verification of the Angles. 

(129.) Kegarding the earth as a sphere, the measured hori- 
zontal angle BAC is plainly the angle formed by tangents to 
two great circles passing through the stations A, B, and A, 
C ; and similarly for every angle measured in the survey. 

These angles are therefore those of spherical triangles 
formed by arcs of great circles intercepted between the 
respective stations (Geom. 448) ; the stations themselves 
being understood to be all referred to the level surface of the 
earth. Hence the sum of the three angles of each of these 
triangles should exceed 180° (Geom. 454). This excess, called 
the spherical excess, will be greater as the triangle is greater, 
and thus becomes a test of the accuracy with which the angles 
have been measured. 

To compute the Spherical Excess. — Compute the area of 
the triangle in square feet, and subtract 9.32677 from the 
logarithm of that area ; the remainder will be the logarithm 
of the number of seconds in the spherical excess. 

This method results from a formula established by Legendke, 
the investigation of which need not here be given. It is 
applicable to spherical triangles which occupy a compara- 
tively small portion of the surface of the earth. 

The computation of the spherical excess, as seen above, 
requires that the area of the spherical triangle shall be known. 
For the use thus made of it, this area is taken equal to that 
of a plane triangle having given parts equal to the measured 
and computed parts of the spherical triangle. 

The difference between the computed excess and that found 
by comparing the sum of the three angles, as measured with 
180°, is the error of measurement. This error is distributed, 
as a correction, among the angles, to make their sum, as it 
should be, equal to 180° plus the computed excess. 



SURVEYING. 139 

" There are few cases in geodesic operations in which the 
spherical excess exceeds 5". In the United States Coast 
Survey, the error in the measurement of the three angles, is 
required, it is said, not to exceed 3". 

Computation of the Sides of the Triangle. 

(130.) The sides of a spherical triangle which are very 
small compared with the radius of the sphere — as is the case 
with those to be computed in a Trigonometrical Survey — 
may be taken for the sides of a plane triangle whose angles 
are equal to the spherical angles each diminished by one- 
third of the spherical excess ; and may thus be computed as 
the sides of a plane triangle. 

The demonstration of this Theorem, which was also dis- 
covered by Legendee, would be out of place among the 
general remarks we are here making. 

To compute the side BC, in the triangle ABC. 

Sine of the angle ACB diminished for spherical excess 

I the measured length of AB 

* \ the sine of BAC diminished for spherical excess 

I the length of BC. 

To compute the side BD, in the triangle BCD. 

Sine of the angle BDC diminished for spherical excess 

: the computed length of BC 

\ \ the sine of BCD diminished for spherical excess 

: the length of BD. 

In like manner, all the remaining sides may be successively 
computed. A side of the last triangle should also be meas- 
ured. If its measured and computed lengths agree, the 
accuracy of all the previous work is confirmed, since the 
computed length is dependent on all the previous operations. 
The side thus measured for proof of the work is called the 
base of verification. 

How near the results of the measurement and the compu- 



140 BOOK III. 

* 

tation of this base should be expected or required to agree, 
cannot be precisely stated ; the difference will usually be in 
proportion to the complexity of the triangulation which has 
been made. " On a French triangulation extending over 
500 miles the difference was less than 2 feet. Eesults of 
equal or greater accuracy are obtained on the U. S. Coast 
Survey." In another French survey, " the base of verifica- 
tion to a series of triangles extending a distance of about 
400 miles, differed less than 12 inches from its computed 
length." 

Latitudes and Longitudes of the Stations. 

(131.) The latitude of a place is the arc of the meridian 
intercepted between that place and the equator; and the 
longitude is the arc of the equator intercepted between the 
meridian of the place and the first or standard meridian — 
which, with the English and Americans, is the meridian of 
Greenwich. 

The geographical position of a place depends on its latitude 
and longitude. These elements are determined for a few of 
the stations, in a Trigonometrical Survey, by means of astro- 
nomical observations. The latitude and longitude having 
been determined, by observation, at any one station, and also 
the bearing or azimuth of another, that is, the angle which 
an arc of a great circle joining the two stations makes with 
a meridian at the first station ; the latitude and longitude of 
the second station may be computed, together with the 
azimuth of a third station as seen from the second ; and 
thus the latitudes and longitudes of the stations might be 
successively determined. The methods to be pursued in 
these computations cannot be here explained ; they depend 
on Spherical Trigonometry. 

Secondary Triangulation. 

(132.) The trigonometrical points, or stations, whose posi- 
tions have been determined, as above, by the primary tri- 
angulation, are taken as points of departure for smaller 



SURVEYING. 141 

triangles within the former, by which other positions are, in 
like manner, determined. 

Within these secondary triangles other still smaller tri- 
angles may be established, if necessary, and thus any desir- 
able number of points or stations may be determined within 
the outlines of the district or country surveyed. 

In the minor details, the common and more expeditious, 
though less accurate, methods of Plane Surveying, with the 
compass and other simple instruments, are employed ; as in 
measuring the minor bearings and distances, tracing the 
course of a road, or stream, or running offsets to a lake, gulf, 
or sea-shore, &c. 

Map of the Survey. 

(133.) From the latitudes and longitudes of stations, and 
other particulars determined by the preceding operations, a 
Map may be projected, representing, as nearly as the case 
will admit, the relative positions of the different parts of the 
district or country surveyed. 

Meridians and parallels of latitude are drawn on a plane, 
at suitable intervals, and the points whose geographical 
positions have been ascertained are put down with reference 
to these lines. Rivers are traced from point to point, also 
mountain ranges, lake shores, sea coasts, &c, and thus the 
map is tilled up to any desirable extent of detail. 

The surface of the earth being nearly spherical, it is 
impossible to represent any considerable portion of it on a 
plane so that the distances between places shall retain the 
same proportions that they have on the sphere ; and hence 
have arisen different methods of delineation, which approxi- 
mately attain the ends in view. 

One of these methods regards the surface surveyed as pro- 
jected on a part of the convex surface of a cone; the two sur- 
faces being conceived to meet each other in the middle paral- 
lel of the surface to be delineated, or to cut each other in two 
parallels equi-distant from the middle one. By this method, 
the maps of states and kingdoms are, for the most part, con- 



142 BOOK III. 

stracted. The convex surface of a cone, when developed, or 
spread out on a plane, forms a sector of a circle whose radius 
is equal to the side of the cone ; and a map thus constructed 
will be a portion of a circular sector, with portions of diverg- 
ing radii for the meridians. 



FIGURE AND MAGNITUDE OF THE EARTH. 

(134.) By the methods of Geodesic or Trigonometrical 
Surveying, measurements have been made from which the 
length of a degree of a meridian, in various latitudes, has been 
computed ; and the figure and magnitude of the entire body 
of the earth have been thence determined. 

The operations necessary for determining,,^/^, the length, 
in linear measure, of a degree of a meridian, and, secondly, 
for determining, from these lengths, as they are found to be 
in different latitudes, the figure and magnitude of the earth, 
are of a very complicated nature, and many of them depend 
on principles which transcend the attainments of those for 
whom this work is intended. We shall therefore only state 
results which are confidently relied on by the most eminent 
mathematicians and astronomers. 

The following are selected from Herschel's Astronomy, in 
which are to be found the computed lengths of a degree of a 
meridian in eighteen different latitudes— the work of as many 
able astronomers, in various countries. The arcs computed 
varied from one to sixteen degrees I we give the length of a 
degree in the latitude of the middle point of the arc. 

In latitude 12° 32' 21", a degree=362956 feet ; 
" 44° 51' 2.5", a degree= 364572 feet ; 

" 52° 32' 17", a degree=365300 feet ; 

" 66° 19' 37' ; , a degree^367Q86 feet. 

The linear measures of the degrees increase contimcally in 
going toward the pole ; a meridional section of the earth 
cannot therefore be a circle, since on the circumference of a 



SURVEYING. 143 

circle the linear measures of the degrees would be equal ; 
hence it also follows that the earth is not perfectly spherical. 

The supposition that a meridional section of the earth is an 
ellipse (Geom. 545) whose minor axis coincides with the axis 
of the earth, agrees with the increasing lengths of the degrees 
from the equator towards the pole ; the major axis of the 
ellipse will then be the equatorial, and the minor the polar 
diameter of the earth. The figure of the earth is therefore 
that of an oblate spheroid (Geom. 568). 

From a combination of thirteen measured arcs, Aery com- 
puted the following dimensions for the earth : 

Equatorial diameter, 7925.648 miles ; 

Polar diameter, 7899.170 miles. 

From a combination of ten measured arcs, which he con- 
sidered the most reliable, Bessel found the dimensions to be 
as follows ; 

Equatorial diameter, 7925.605 miles ; 

Polar diameter, 7899.1 14 miles. 

These last are the most recent determinations of the magni- 
tude of the earth, and are regarded by many eminent astro- 
nomers as the most correct ; but the differences between these 
and the former are less than one tenth of a mile. 

The earth's mean diameter, half the sum of the greatest 
and least, is 7912.359 miles ; the excess of the equatorial 
above the polar diameter is 26.49 miles ; this excess is about 
3-^0 of the longest diameter, and is called the ellipticity of 
the earth. 

This proportional ellipticity is so small that, in any correct 
model of the earth, it would be imperceptible otherwise than 
by accurate measurements. In a terrestrial globe represent- 
ing the spheroidal figure, if the equatorial diameter be 12 
inches, the polar diameter must be about 11.95 inches. For 
most practical purposes involving a consideration of the 
curvature of the earth's surface, it may be regarded as per- 
fectly spherical. 



7*. 



BOOK IV. 



NAVIGATION. 



(135.) Navigation is the art of conducting a ship on the 
ocean ; and has special reference to the determination, at any 
time, of her latitude and longitude. 

In Navigation, the surface of the ocean is regarded as per- 
fectly spherical y so that every intersection of that surface by 
a plane passing through the centre of the earth is the circum- 
ference of a great circle. 



Latitude and Longitude. 

(136.) The Latitude of a place is the arc of a meridian 
intercepted between that place and the equator. It is called 
north latitude, or south latitude, according as the place is 
north, or south, of the equator ; and can never exceed 90° o 



Let EQB- be the equator, which 
is a great circle having E" and S for 
its poles ; and let ScN be a meridi- 
an (76) passing through the points 
h and d ; then the arc cb is the la- 
titude of the point h, and cd is the 
latitude of the point d. The first 
of these points is in north latitude, 
the second is in south latitude. 




Parallels of latitude, as dbh and dmo, are the circumfer- 
ences of small circles of the earth parallel to the equator. 

The difference of latitude of two places is the arc of a 
meridian intercepted between the parallels of latitude passing 
through the two places. It is found by subtracting the less 



NAVIGATION. 145 

latitude from the greater when the two places are on the same 
side of the equator, and by adding the two latitudes together 
when the places are on opposite sides of the equator. 

Thus the arc bd is the difference of latitude of the points b 
and d, and also of the points b and m. 

(137.) The Longitude of a place is the arc of the equator 
intercepted between the meridian of that place and the first 
meridian. It is called east longitude, or west longitude, 
according as the place is east, or west, of the first meridian; 
and can never exceed 180°. 

By the first meridian is meant that one from which it has 
been agreed that the longitude of places shall be reckoned. 

The English reckon longitude from the meridian of Green- 
wich Observatory, in the vicinity of London, and the same 
custom has prevailed in the United States. 

In the preceding figure let SEN be the first meridian, and 
ScN a meridian passing through the points b, c, and d ; then 
the arc Ec, of the equator, is the longitude of each of these 
points, and this longitude is east, since these points lie east 
of the first meridian. The arc E5 is the longitude, west, of 
the point b. 

The Difference of longitude of two places is the arc of the 
equator intercepted between the meridians of the two places. 
It is found by subtracting the less longitude from the greater 
when the two places are on the same side of the first meridian, 
and by adding the two longitudes together when the places 
are on opposite sides of the first meridian ; but when the sum 
of the opposite longitudes exceeds 180°, the difference of 
longitude will be found by subtracting that sum from 360°. 

The are ex is the difference of longitude of the points c 
and x / also of the points b and m, &c. 

Ship's Course. 

(13§.) The Course of a ship, at any time, is the angle 
which her track makes with the meridian she is crossing; 



146 



BOOK IV. 



arid a ship is said to continue on the sa?ne course when her 
track everywhere cuts the meridian at the same angle. 

The surface of the ocean being spherical, and the meridians 
not parallel, the track thus described by the ship is not a 
straight line, nor a plane curve except when the ship sails 
on the equator, a meridian, or a parallel of latitude. It is a 
kind of spiral, and is called a rhumb line or loxodromic curve. 

The ship's course is 
shown by the Mariner's 
Compass,\vh.ich contains 
a circular card attached 
to a magnetic needle ba- 
lanced on a pivot, so as 
to move freely in a hori- 
zontal plane. The nee- 
dle is fastened to the un- 
der side of the card, in 
the direction of the dia- 
meter NS, which there- 
fore indicates the mag- 
netic meridian. 

The circumference of the card is divided into thirty-two 
equal parts, called rhumbs or points, and each of these is 
subdivided into four equal parts, called quarter-points. 

The cardinal points of the compass are, the north, south, 
east, and west points. The point midway between north and 
east is called north-east, the one midway between south and 
east is called south-east, &c. The other points are also 
named in a regular manner; thus, beginning at north and 
going around the compass by east, the thirty-two points are: 

North, north by east, north-north-east, north-east by north, 
north-east, north-east by east, and so on, as marked in the 
figure. 

The interval between two adjacent points is 

360°H-32=lli degrees, 
A quarter-point is one fourth of this interval, that is, 
Hi-7-4=2i§ degrees=2° 48' 45". 




NAVIGATION 



147 




Shifts Bate of Sailing. 

(139.) The rate at which the ship runs is determined by 
the log-line. 

This consists of a line attached to a piece 
of wood having the form of a sector of a 
circle, with its rim loaded with lead, to 
maintain it in a vertical position, when 
thrown upon the water. 

The line is attached at three points on the broad side of 
the log, which turns that side towards the ship, and thus it 
is prevented from being drawn 
along in the water. The ship's 
progress is then ascertained by 
the rate at which the line un- 
winds from a reel on board. 

For convenience the line is divided into equal parts, called 
knots, each of which is the 120th part of a mile ; so that the 
number of knots unwound in half a minute is the number of 
miles the ship is running per hour. 

The half-minute is measured by a sand-glass, 
consisting of two parts united by a neck, through 
which a quantity of sand runs in that time. 

The mile by which distances are expressed in 
Navigation is the nautical or geographical mile ; 
and is equal to one minute, or the 60th part of 
a degree, on the circumference of a great circle 
of the earth. 





The Log-hook is a record kept of all matters relating to the 
navigation of the ship ; among which are the course and rate 
of sailing at different hours of the day, the latitude and lon- 
gitude of the ship at noon, the course and distance made, on 
the whole, during the day. This continuous record is the 
Ship's Journal. 



148 



BOOK IV 



PLANE SAILING. 



(140.) Plane Sailing is a general designation of the 
methods of solving problems in Navigation in which are 
involved the ship's course, distance run, difference of latitude, 
and departure, — by means of right-angled plane triangles. 



Let EQ be a portion of the equator, P the 
pole of the earth, and AB a ship's trade on 
a uniform course, that is, cutting all the 
meridians at the same angle. 

Let AB be divided into parts, Ab, bo, &c, 
so small that each of them may be con- 
sidered a straight line ; through the points 
A, b, c, &c, draw meridians, and to these 
meridians the perpendiculars be, cf, &c. 




It is plain Ae+bf+cg+dh is the whole distance that the 
ship has advanced northward, measured on a meridian ; and 
that eb+fc+gd+fiB is the whole distance that she has 
advanced eastward, measured perpendicularly to a meridian. 



If, therefore, in the right-angled plane tri- c J tyarcare. 
angle ACB, the angle A be made equal to 
the ship's course, that is, the angle at which 
the ship's track everywhere cuts the meridi- 
an ; AB equal to the distance run ; and AC 
equal to the sum of the minute northings Ae, 
bf, &c, on a meridian; the perpendicular 
CB will be equal to the sum of the minute 
eastings eb, fc, &c, perpendicular to the me- 
ridians, that is, equal to the whole departure regarded as the 
sum of all the minute departures from the successive 
meridians. 

The following principles are therefore deducible : 

(141.) 1. The departure which a ship, on a uniform course, 
makes with reference to the meridian sailed from, is the sum 




NAVIGATION. 149 

of all her minute departures from the meridians she is con- 
tinually crossing. 

2. If a right-angled triangle be formed with an acute angle 
equal to the ship's course, and the hypothenuse equal to the 
distance run; the other side adjacent to that angle will be 
the difference of latitude, and the side opposite will be the 
departure, made. 

The solution of problems in Plane Sailing is thus reduced 
to the solution of right-angled plane triangles. Any two of 
the four elements, course, distance, difference of latitude, and 
departure, being given ; the other two may be readily com- 
puted (54). 

Traverse Sailing. 

(142.) A Traverse is the track described by a ship in 
sailing from one point to another by different courses / and 
the working of the traferse consists in determining the single 
course and distance from one of those points to the other. 

The difference between the sums of the northings and 
southings of the given courses and distances will be the 
northing, or southing, and the difference between the sums 
of the eastings and westings will be the easting, or westing, 
of the required single course and distance. Having thus 
found the difference of latitude and the departure, the course 
and distance may be computed from a right-angled triangle 
(141...2). 

These operations proceed on the principles of Plane Sailing, 
and the problem to be solved is evidently like that of finding 
the bearing and length of one side of a survey, when the 
bearings and lengths of the other sides are given (88). 



In the preceding method of working or compounding a 
Traverse, it is assumed that, when a ship sails from one point 
to another by several courses, she makes the same departure 
as if she had proceeded by a single course to the same point. 



150 



BOOK IV. 



This is not strictly true, since — the meridians being con- 
vergent towards the pole — the minute constituent departures 
on different tracks between the same two points must differ. 

The resulting single course and distance will be most cor- 
rectly found when the Traverse is in the vicinity of the 
equator, because the meridians are there most nearly parallel. 
In general, the result is considered sufficiently accurate for a 
traverse not exceeding twenty-four hours' continuance. 



EXAMPLE. 



A ship from latitude 47° 30' K has sailed S. W. by S. 98 
miles. What latitude is she in ? and what departure has she 
made ? 

The course, south-west by south, is 3 points from south, 

equal to 

111° X 3=33° 45'. 

In the right-angled triangle ACB let the 
acute angle A be equal to the course, 33° 45', 
and the hypothenuse AB equal to the dis- 
tance 98; then AC will be the difference 
of latitude, and BC the departure, made 
(141 ..2). 

To find AC (44). 

Eadius 10 10 10.000000 

is to AB 98 1.991226 

as cos A 33° 45' 9.919846 

is to AC 81.48 .' 1.911072 




"We thus find the difference of latitude the ship has made 
to be 81.48 miles. These miles are so many minutes on a 
meridian ; and are therefore equal to 

81.48-7-60=1° 21.48'. 



NAVIGATION. 151 

The ship having sailed from north towards south, the lati- 
tude arrived at will be found by subtracting the difference 
of latitude made from her original latitude 

(47 30')-(l° 21.48')=46° 8.52'. 

The ship therefore arrived at 46° 8.52' north latitude. 

To find BC (44). 

Eadius 10 10 10.000000 

is to AB 98 1.991226 

as sin A 33° 45' 9.744739 

is to BC 54.44 1.735965 

The ship has therefore made 54.44 miles of west departure. 

1. A ship sails from latitude 38° 31' ~N., on a course which 
runs" S. 43° 20' E., the distance of 132 miles. At what lati- 
tude has she arrived ? and what departure has she made ? 

Ans. Latitude 36° 55' 1ST. ; departure 90.58 miles E. 

2. A ship sails from latitude 48° 27' S., on a course which 
runs S. W. by "W., at the rate of 7 miles an hour. In what 
time will she arrive at the parallel of 50° S. latitude ? 

Ans. 23.91 hours. 

3. If after a ship has sailed from latitude 40° 21' N. to 
latitude 46° 18' N"., she be found to have made 216 miles of 
east departure ; what was her course ? and what distance 
did she make? 

Ans. K 31° 11' E. ; distance 417.25 miles. 

4. A ship sails from latitude 3° 52' S. to latitude 4° 30' K, 
on a course which runs ]ST. "W". by W \ W. Kequired the 
distance and departure. 

The expression for the course in this problem signifies that 
the course is half a point farther from E". than N. W. by W. ; 
that is, 5 J points from the meridian. 

Ans. Distance 1065.2 miles ; departure 939.5 miles. 



152 BOOK I Y. 

5. Two ports lie on the same meridian, one in latitude 52° 
30' E., the other in latitude 47° 10' E. A ship sails from 
the southernmost port due east at the rate of 9 miles an hour, 
and in two days meets a sloop which had sailed from the 
northernmost port. What was the sloop's direct course? 
and the distance run ? 

Ans. S. 53° 28' E. ; distance 537.6 miles. 



6. A ship's progress for twenty-four hours is found to have 
been as follows : S. E., 40 miles ; 1ST. E., 28 miles ; S. "W. by 
W., 52 miles ; E. "W. by W., 30 miles ; S. S. E., 36 miles; 
S. E. by E., 58 miles. Find the direct course and the distance 
from the point left to that arrived at. 

This is a problem in Traverse Sailing. In order to its 
solution it will be expedient to form a Table containing 
columns for the courses, distances, &c, as in the correspond- 
ing case in Surveying (88). 

Ans. S. 25° 56' E. ; distance 95.80 miles. 

7. A ship from latitude 28° 32' E., has run the following 
courses, viz. : E. W. by E., 20 miles ; S. W., 40 miles ; E. 
E. by E., 60 miles ; S. E., 55 miles ; W. by S., 41 miles ; E. 
E. E., 66 miles. Required the direct course and distance 
from the place left to that arrived at. 

Ans. E. 89° 58' E. ; 70.16 miles. 

8. A ship from Bermuda, in latitude 32° 22' E. ran 1ST. E., 
40 miles, then E. E. by E., 20 miles, then E. E. by E., 50 
miles, than E., 38 miles, then E. E., 10 miles, then E. E. E., 
60 miles. Required her present latitude, the distance made 
good, and the direct course from Bermuda to the point 
arrived at. Ans. 34° 37' ; 201.18 miles; E. 47° 47' E. 



9. A ship sailed S. by W. for 5 hours, at the rate of 7 miles 
an hour, as shown by the compass and log-line, but in a 



NAVIGATION. 



153 



current of the ocean which set "W". by 1ST., at the rate of 3 
miles an hour. What was the true course on which she 
sailed ; and the distance made on that course ? 

The effect of the current was the same that would have 
resulted, in still water, from an additional course in the 
same direction, with the same rate of motion as the current. 
The problem may therefore be solved by the method of 
Traverse Sailing. Ans. S. 34° 27' W. ; 38.07 miles. 

10. A ship's Traverse, as shown by the compass and log- 
line, was as follows, viz. : E. by S., for 2 hours, at 6 miles an 
hour ; E. S. E., for 3 hours, at 7 miles an hour; S. E. by E., 
for 4 hours, at 8 miles an hour. But during the whole time 
she was in a current setting E. by JST., at the rate of 2 miles 
an hour ; what was the direct course of the ship ? and the 
distance made on that course ? 

Ans. S. 71 54'E.: 79.34 miles. 



PARALLEL SAILING 

(143.) Parallel Sailing designates the case of a ship 
sailing on a parallel of latitude ; in which, from the latitude, 
and the distance sailed, may be readily determined the differ- 
ence of longitude made. For, 

The cosine of the latitude of any parallel is to radius, as 
any distance on that parallel is to an arc of the equator inter- 
cepted between the same meridians, that is, 

The Difference of Longitude. 

Let C be the centre of the earth, 
P the pole of the equator EQE, 
and AB the distance sailed on a 
parallel of latitude ; also let PAQ 
and PBE be two meridians pass- 
ing through the points A and B, 
and intercepting the arc QE of the equator. 




154: 



BOOK IV. 




AQ is the latitude of the parallel 
AB (136), and QK is the differ- 
ence of longitude of the points A 
and B (137) ; also FA is the sine 
of the arc AP, or cosine of AQ. 
Moreover FA is the radius of the 
parallel AB, and CQ is the ra- 
dius of the equator, and also of the meridian PAQ ; so that 

FA : CQ : : AB : QE (Geom. 258) ; 

or, the cosine of the latitude of the parallel is to the radius, 
as the distance sailed on that parallel is to the difference of 
longitude made. 



(144.) If a right-angled triangle be formed with one of its 
perpendicular sides, AB, equal to the distance sailed on a 
parallel of latitude, and the adjacent acute angle, B, equal to 
the latitude; the hypothenuse, BC, will be equal to the 
difference of longitude made. 



For, radius : BC \ \ cos B : AB (44), c 

or AB . radius=cos B . BC ; 

or cos B : radius ; : AB : BC (Geom. 
180); 




distance 



and these are the same relations that have already been esta- 
blished for the latitude, distance, and difference of longitude 
in Parallel Sailing. 



Middle Latitude Sailing. 

(145.) Middle Latitude Sailing designates a method of 
finding the difference of longitude which a ship makes on an 
oblique course, by considering her departure equal to the 
distance— on the parallel of middle latitude— between the 
meridian sailed from and the one arrived at. 



NAVIGATION. 



155 




Suppose the ship's track to be AB ; and 
let EF be a parallel of latitude midway be- 
tween the parallels AC and DB ; then it is 
assumed that the departure (141...1), which 
the ship makes in going from A to B is 
equal to the distance EF between the meri- 
dians PA and PBC. 



The middle latitude is equal to half the sum, of the two 
extreme latitudes when those two latitudes are both north, 
or both south ; and equal to half their difference when one 
is north and the other south. 

On these principles we have therefore the following rule. 

(146.) To find the difference of longitude which a ship 
makes on an oblique course.— Find the departure by the 
method of Plane Sailing (141...2) ; consider that departure as 
distance made on the parallel of middle latitude, and then 
find the difference of longitude as in Parallel Sailing (143). 

Any problem in Middle Latitude Sailing may be repre- 
sented by a combination of two right-angled plane triangles. 
Thus : 

(147.) In a right-angled triangle ACB, let the angle A be 
equal to the ship's course, the hypothenuse AB equal to the 
distance sailed, and the perpendicular BC equal to the 
departure made (141. ..1) ; then, 

Make the angle CBD equal to the middle latitude ; com- 
plete the right-angled triangle BCD, and the hypothenuse 
BD will be the difference of longitude made. 

This follows from considering that, 
on the principles of Middle Latitude 
Sailing, the diff. of long, made is the 
same as if the ship's course had been 
on the parallel of middle latitude, and 
her distance on that course equal to 
the departure made on her oblique 
course (145). 




156 BOOK IV. 

From these triangles we derive the following proportions, 
which may be applied to different cases of a problem in 
Middle Latitude Sailing. 

(14§.) 1. The cosine of the middle latitude is to the 
departure, as radius is to the difference of longitude. 

This is seen in the triangle BCD — recollecting that the 
the angle CBD is equal to the middle latitude. 

2. The cosine of the middle latitude is to the distance 
sailed, as the sine of the course is to the difference of longi- 
tude. 

This is seen in the triangle ABD — recollecting that the 
angle D is the complement of CBD, that is, of the middle 
latitude. 



3. The cosine of the middle latitude is to the difference of 
latitude as the tangent of the course is to the difference of 
longitude. 

This follows from the first proportion above stated, and 
the proportion 

AC : radius ; : BC : tang. A (46). 

For by comparing the products of the extremes and means 
in the two proportions, we shall find cos CBD . BD=radius 
. BC=AC . tang. A, which gives 

cos CBD : AC : : tang. A \ BD. 



Correction of the Middle Latitude Method. 

(149.) The principle assumed in Middle Latitude Sailing 
— that the departure made on an oblique course is equal to 
the distance on the parallel of middle latitude between the 
meridian sailed from and the one arrived at — is not strictly 
correct; it is, indeed, quite erroneous when the course is a 
small angle, or the distance run is great. For, in either of 
these cases — as will appear from inspecting the figure in 



NAVIGATION. 157 

(145), — the middle latitude distance will receive greater 
accessions than the departure, as the ship proceeds on her 
course. But ' 

The results of this method may, in all cases, be rendered 
accurate, by applying to the middle latitude a correction from 
Table IY., — the construction of which will be explained here- 
after. 

EXAMPLE. 

A ship, in latitude 51° 18' K, longitude 22° 6' W., sailed 
S. 33° 5' E., the distance of 1024 miles. What difference of 
longitude did she make ? and to what longitude did she sail ? 

The difference of longitude is to be obtained by means of 
the middle latitude, and to find this we must first have the 
latitude arrived at. 

In the right-angled triangle ABC (147), 

Kadius 10 10 comp. log. 0.000000 

is to AB 1024 3.010300 

as cos A 33° 5' 9.923181 

is to AC 857.98 2.933481 

The difference of latitude is thus found to be 857.98 miles. 
Taking the nearest integral number of miles or minutes, we 
obtain 

858-^60=14° 18'. 

The ship therefore sailed to (51° 18 / )-(14° 18')=37° 1ST. 
latitude. 

Hence the middle latitude, which in this case is half the 
sum of the two extreme latitudes, is 

\ (51° 18 / +37°)=44° 9'. 

We have now to add to this middle latitude the correction 
corresponding to it, for a difference of latitude of 14° 18'. 
Taking the nearest number of degrees in Table IY., opposite 
to 44°, and under 14°, we find 25' ; then 

44° 9'+ 25'=44° 34'. 



158 BOOK IV. 

We are now prepared to find the difference of longitude ; 
thus, 

cos. corrected mid. lat. 44° 34' . . comp. log. 0.147255 

is to the distance 1024 3.010300 

as sin. course 35° 5' 9.759492 

is to diff. long. 826.12 2.917047 

(148...2). 

We have thus found the difference of longitude to be 
826.16 miles or mimites. Taking the nearest integral number, 
we obtain 

826^60=13° 46'. 

To find the longitude sailed to. — The ship sailed from 
longitude 22° 6' W., towards the south and east ; hence the 
longitude arrived at was 

22° 6'-(13° 46')=8° 20', west 

11. A ship from latitude 53° 56' K, longitude 10° 18 / E., 
has sailed due west, 236 miles. What is her present longi- 
tude ? 

First find the difference of longitude the ship has made, 
(143). Ans. Longitude 3° 37' E. 

12. A ship in latitude 32° E". sails due east until her 
difference of longitude is 6° 24'. How many miles has the 
ship sailed ? 

The difference of longitude must first be reduced to miles. 

Ans. Distance run, 325.6 miles. 

13. If a ship in latitude 49° 30' K or S. should sail directly 
west 136.4 miles, what difference of longitude would she 
make? Ans. 3° 30'. 

14. A ship leaving a port whose latitude is 38° N"., and 
longitude 16° E., sails west on a parallel of latitude 117 miles 
in 24 hours. What difference of longitude does she make ? 
and what is her longitude at the end of the time ? 

Ans. 2° 28', and 13° 32 / E. 



NAVIGATION. 159 

15. A ship sailed from latitude 42° 30' K, and longitude 
58° 51' W., on a course which ran S. E. by S., till the distance 
made was 300 miles. Find the latitude and the longitude at 
which the ship arrived. 

Problems in Middle Latitude Sailing will be most readily 
solved by a reference to the combination of triangles in (147). 
Thus, for this problem we shall have the angle A, which is 
the given course, and the side AB, which is the given 
distance. 

Find the difference of latitude, AC, which, in this problem, 
will be south, and when applied to the latitude from which 
the ship sailed, will show the latitude reached. Find the 
departure BC, from which and the middle latitude, or angle 
CBD, find the difference of longitude BD. This difference 
applied to the longitude from which the ship sailed will show 
the longitude reached. Ans. 37° 21' K ; 55° 13' W. 

16. A ship sails, in the N". W. quarter, 248 miles, when her 
departure is found to be 135 miles, and her difference of 
longitude 310 miles. Find the direct course on which she 
sailed. Ans. K 32° 59' W. 

17. A ship from latitude 37° K, longitude 9° 2' W., having 
sailed between 1ST. and W. 1027 miles, finds that she has 
made 564 miles of departure. What was her direct course % 
and the longitude reached? 

Ans. K 33° 19' W. ; and 22° 14' W. 

18. Find the course, and the distance, on a rhumb line, 
from New York, in latitude 40° 42' 1ST., longitude 74° V W., 
to Liverpool, in latitude 53° 22' K, longitude 2° 52' W. 

Ans. K 75°l6' E. ; 2988.3 miles. 

19. Find the course, and the distance, on a rhumb line 
extending along the level surface of the earth, from the city 
of Washington, in latitude 38° 58' K, longitude 77° 2' W., 
to Boston, in latitude 42° 22' K, longitude 71° 4' W. 

Ans. N. 53° 5' E. ; 339.55 miles. 



160 



BOOK IV. 



MERCATOR'S SAILING. 

(150.) Mekcator's Sailing designates a method of finding 
the difference of longitude which a ship makes on an oblique 
course, by proportionally increasing her difference of latitude 
and departure, so that the latter may become equal to an 
arc of the equator intercepted between the meridian sailed 
from and the one arrived at. 

The right-angled triangle ACB repre- 
sents all the elements which are involved 
in the problem of a ship's sailing on an 
oblique course, or rhumb line, excepting 
the difference of longitude (141. ..2). 

In Mercator's Sailing, the difference of 
latitude, AC, and the departure, CB, are 
supposed to be increased, proportionally, 
thus becoming AD and DF,— DF being 
equal to an arc of the equator intercepted 
between the meridians passing through 
the extremities of the ship's track, and therefore equal to the 
difference of longitude made. 

The increased difference of latitude, AD, is called the 
meridional difference of latitude, to distinguish it from the 
true difference of latitude, AC. 

The two right-angled triangles ACB and ADF are similar, 
and we have the proportions : 




(151.) 1. The true difference of latitude is to the merid- 
ional difference of latitude, as the departure is to the differ- 
ence of longitude. 

Thus AC : ad : : cb : df. 

2. Eadius is to the tangent of the course, as the meridional 
difference of latitude is to the difference of longitude. 

Thus E : tang. A : : AD : DF (46). 

Other proportions may readily be derived from the same 
triangles. 



NAVIGATION. 161 



TABLE OF MEKIDIONAL PARTS. 

(152.) Table Y, appended to this work, contains the 
increased latitudes, or parts of a meridian, expressed in 
nautical miles, from which the meridional difference of lati- 
tude is to be obtained, when the ship's departure is increased 
to her difference of longitude. 

Thus for latitude 40° 30' we find the number 2622 ; 
and for latitude 46° 10' " " 3130. 

"When these latitudes are both north, or both south, the 
meridional difference of latitude will be obtained by sub- 
tracting the less number from the greater ; when one of the 
latitudes is north and the other south, the meridional differ- 
ence of latitude will be obtained by adding the two numbers 
together. 

Computation of the Meridional Parts. 

(153.) The ship's departure is increased to her difference 
of longitude by increasing each minute of departure from the 
meridians she is continually crossing, to a nautical mile, 
which is 1' on the equator. It is required to find the propor- 
tional increase of the minutes of the meridian in the same 
latitudes with these minutes of departure. 

It has been shown that the 

cos. of the lat. : E ! : V of dep. in that lat. :V of 'the equator, 

(143). 

But 1' of departure, in the present theory, is 1 mile ; and 
1' of the equator is equal to 1' of a meridian ; hence 

cos. of lat. I R \ \ 1 mile \V of a meridian. 

The ratio between the cosine and the radius remains 
unchanged when the latter is made unity, that is, 1 mile. 
Hence 

nat. cos. of lat. 1 1 mile ; *. 1 mile \Y of a meridian ; 



162 BOOK IT. 

which gives V of a meridian — — „ 7 , = nat. sec. of 

b J nat. cos. of lat. J 

lat. (26...2). 

It has thus been shown that, in the theory of Mercator's 
Sailing, V of a meridian, taken in any latitude, becomes 
equal to the natural secant of the latitude. 

Then, beginning at the equator, 

the 1st minute of the meridian=nat. sec. 1' ; 
2d minute of the meridian = nat. sec. 2' ; 
Sd minute of the meridian=nat. sec. 3', &c. 

The Table of Meridional Parts is formed by the continual 
addition of these separate results — as found in a Table of 
Natural Secants, 

merid. parts of l'=nat. sec. 1'; 

merid. parts of 2'=nat. sec. l'+nat. sec. 2'; 

merid. parts of 3'=nat. sec. l'+nat. sec. 2'+nat. sec. 3', &c. 

Mercator's Chart is one on which the meridians of the 
earth are represented by parallel straight lines, with each 
minute of latitude increased in the proportion of the secant 
to the radius, as in the theory of Mercator's Sailing. On 
this Chart a rhumb line becomes a straight line cutting the 
parallel meridians at equal angles. 

EXAMPLE. 

Find the course and distance from Cape Cod Light-house, 
in latitude 42° 3' K, longitude 70° 4' W., to the Island of 
St. Mary, in latitude 36° 59' K, longitude 25° 10' W. 

Latitudes, 42° 3' N* Meridional parts, 2785.8 K 

36° 59 7 K 2391.4 K 

Diff. of lat, 5° 4'= 304 m. Merid. diff. of lat. 394.4 miles. 
Diff. of long. 44° 54'=2694 miles. 

To find the Course. — In the right-angled triangle ADF 
(150), we shall have the meridional difference of latitude, 



NAVIGATION. » 163 

AD, 394.4 miles, and the difference of longitude, DF, 2694 
miles, to find the angle A. 

AD 394.4 comp. log. 7.404063 

is to radius 10 10 10.000000 

as DF 2694 3.430398 

is to tang. A 81° 40' 10.834461 (46). 

Cape Cod being in higher latitude than St. Mary, and 
further west in longitude, the course from the former to the 
latter place is thus found to be 

S. 81° 40' E. 

To find the Distance. — In the right-angled triangle ACB 
(150), we have the true difference of latitude, AC, 304 miles, 
and the angle A 81° 40', to find the distance AB, as in Plane 
Sailing. AB will be found to be 2097.5 miles. 

20. Find the course and distance from a port in latitude 
37° 48' K, longitude 25° 13' W., to another which lies in 
latitude 50° 13' K, longitude 3° 38' W. 

Am. K 51° 11' E. ; distance 1188.51 miles. 

21. A ship sailed from Havana, in latitude 23° 9' E"., 
longitude 82° 19' W., on a course which ran N". E. by !N"., 
until the distance made was 500 miles. What latitude and 
longitude did the ship reach ? Ans. 30° 5' K ; 77° 8' W. 

22. On what uniform course must a ship sail from Cape 
Horn, in latitude 55° 58' S., longitude 67° 21' W., to arrive 
at Havre-de-Grace, in latitude 49° 29' K, lougitude 0° 6' E., 
and what distance must she make ? 

Ans. K 28° 22' E. ; 7190.3 miles. 

23. A ship sailed from St. Augustine, in latitude 29° 52' 
R, longitude 81° 25' "W., in the K E. quarter, to latitude 
34° 40', and the distance made was estimated at 520 miles. 
"What longitude did she reach? Ans. 72° 53' W. 



164 ' BOOK IV. 

24. A ship in latitude 42° 30' K and longitude 58° 51' W., 
sails S. W. by S. 300 miles. Required the latitude and longi- 
tude arrived at. Ana. 38° 21' K ; and 62° 29' W. 

25. On what course, continued invariably, must a ship 
sail from the Bay of San Francisco, in latitude 30° 23' JST., 
longitude 115° 36' W., to reach Canton, in latitude 23° 7' K, 
longitude 113° 14' E., and what would be the distance 
sailed ? 

In this problem the difference of longitude of the two 
places will be found by subtracting the sum of their longi- 
tudes from 360°, since the two longitudes are on opposite 
sides of the first meridian, and their sum exceeds 180°. 

Ans. S. 86° 27' W. ; 7041.38 miles. 



Correction of Middle Latitude. 

(1541.) It was assumed, in Middle Latitude Sailing, that 
when the middle latitude, found from the two extreme lati- 
tudes, is increased by the proper quantity from Table IV., a 
latitude will be obtained in which the distance between the 
meridian sailed from and the one arrived at is accurately 
equal to the departure made (149). 

We are now to establish a formula by which these correc- 
tions of the middle latitudes may be computed — which, as 
will be seen, could not be done prior to the consideration of 
Mercator's Sailing, since the procedure requires the merid- 
ional difference of latitude. 

Let d represent the true diff. of lat, D the meridional diff. 
of lat., m the middle lat., Mihe corrected middle lat., and L 
the diff. of long. 

Then, M being the latitude in which the distance between 
the meridian sailed from and the one arrived at is equal to 
the departure, we have 

cos M : d ! ! tang, course : L (148.. .3) ; 

. cos. M . L 
which gives tang, course = 



NAVIGATION. 165 

We have also E : tang, course ! I D : L (151...2) ; 

which gives tang, course = • 

By equating the two values of tang, course we shall find 



cos M 



D 



But M is the middle latitude, m, increased by the required 
correction', then 

cos (m+the correction) = . 

This last equation shows that (m+the correction) is equal 
to an arc whose cosine is R . d-r-T) ; so that, by transposing 
m, we find 

. . R.d 

the correction = an arc whose cosine is — — m. 

From this formula the corrections contained in Table IV. 
may be computed. Assume unity for the Radius ; then d-r- 
D will give the natural cosine of the correction ; find the 
corresponding number of degrees and minutes in a table of 
natural cosines ; subtract the middle latitude, m, and the 
remainder will be the correction. 



Uncertainty of the Ship's Place as determined oy Dead 



(155.) What is called the ship's place by dead reckoning 
is her latitude and longitude, as computed from the observed 
course and distance that she has sailed from a place whose 
latitude and longitude are known. 

If the course and distance could always be accurately 
determined, the place of the ship could be computed with 
corresponding exactness, by the methods which have been 
explained. But the course and distance can be obtained 
only in a roughly approximative form — which, with the 



166 BOOK IT. 

effect of unknown currents, unavoidable imperfections in 
steering, and numberless other sources of error, renders the 
place of the ship, as estimated from the reckoning, very 
doubtful. 

In order therefore, to determine her place with that pre- 
cision and certainty which the safety of navigation requires, 
recourse must be had to methods entirely independent of the 
dead reckoning. These methods deduce the latitude and 
longitude the ship is in from celestial observations, with the 
aid of Spherical Trigonometry / and the systematic exposi- 
tion of these methods constitutes that branch of Navigation 
which goes under the name of Nautical Astronomy. 

"In the modern practice of Navigation, the course and 
distance are used only to enable the seaman to assign approx- 
imately the place of the ship between the times at which it 
is determined, independently, by celestial observations." 



BOOK V. 

SPHERICAL TRIGONOMETRY, 

AND ITS MOEE IMMEDIATE APPLICATIONS, 



(156.) Spheeical Trigonometry is that branch of Mathe- 
matics which treats of the relations among the sides and 
angles of spherical triangles. 

Its more immediate and practical object is, to establish the 
methods by which, when any three of the six parts of a 
spherical triangle are given, the other three may be com- 
puted. 

THEOREM I. 

(157.) In every right-angled spherical triangle* Radius is 
to the sine of the hypothenuse, as the sine of either of the two 
angles adjacent to the hypothenuse is to the sine of the oppo- 
site side. 



Let ABC be a triangle, right- 
angled at B, on the surface of 
a sphere whose centre is ; 
then 

E : sin AC : : sin B AC : sin BC. 



Draw the straight lines OA, 
OB, and OC ; draw CD perpendicular to OA, and DF, in the 
plane ABO, also perpendicular to OA ; also draw the straight 
line FC. 

Because the straight line AO is perpendicular to DC and 
DF, it is perpendicular to the plane DFC (Geom. 322) ; and 
therefore the plane ABO, passing through the straight line 
AO, is perpendicular to DCF (Geom. 328). But the plane 
BCO is also perpendicular to ABO, since the angle ABC is 
a right angle (Geom. 432) ; hence CF, the common section 

8* 




168 ■ BOOK V. 

of the planes DFC and BOC, is perpendicular to the plane 
ABO (G-eom. 329), and therefore perpendicular to the straight 
lines DF and OB (Geom. 315). 

We have therefore CD for the sine of the hypothenuse AC, 
and CF for the sine of BC (16). 

THe right-angled plane triangle DFC gives 

E : DC : : sin CDF : CF (44). 

And since the angle CDF measures the diedral angle 
formed by the planes ACO and ABO (Geom. 336), which is 
the same as the spherical angle BAC — and DC and CF are 
the sines of AC and BC — we have 

K : sin AC ! : sin BAC : sin BC. 

Therefore, in every right-angled spherical triangle, &c. 

theorem n. 

(158.) In every right-angled spherical triangle, Radius is 
to the cosine of either of the sides containing the right angle, 
as the cosine of the other is to the cosine of the hypothenuse. 



Let the spherical triangle ABC f \x n 



be right-angled at B ; then 

K : cos bc : : cos ab : cos AC. 



--D 



Produce each side of this triangle „/ j 

until it meets the arc DF of a great a^-^ / 

circle described from A as its pole ^^^^^__b/ 
(Geom. 431). 

The great circle ABD, passing through the pole A of the 
great circle DGF, is a secondary to the latter circle, and the 
angles ADF and AGF are right angles (Geom. 442) ; also, 
ABF and ADF being right angles, the arcs BF and DF 
meet in the pole F of the arc AD. 

The arcs ABD,- ACG, and BCF are therefore quadrants 
(Geom. 440) ; so that BD is the complement of AB; CG is the 
complement of AC, and CF is the complement of BC. 



SPHERICAL TRIGONOMETRY. 



169 



In the triangle CGF, right-angled at G, we have 

E : sin CF : : sin F : sin CG (157). 

But the sine of the angle F is the same as the sine of the 
arc BD, which measures that angle (Geom. 447) ; and con- 
sidering now that the sine of an arc is the cosine of the com- 
plement of that arc, we have, 

r : cos bc : : cos ab : cos AC. 

Therefore, in every right-angled spherical triangle, &c. 

(159.) Cor. — In every right-angled spherical triangle, 
Radius is to the cosine of either of the sides containing the 
right angle, as the sine of the angle opposite the other is to 
the cosine of that opposite the first side. 

For in the right-angled triangle CGF, we have 

R I sin CF ; : sin C : : sin GF (157). 

Considering that CF is the complement of BC, that the 
vertical angles at C are equal (Geom. 450), and that GF is 
the complement of GD, which measures the angle A, we 
have, in the triangle ABC, 

R : cos BC ; : sin C :• cos A. 



THEOREM III. 

(160.) In every right-angled spherical triangle, Radius is 
to the sine of either of the sides containing the right angle, as 
the tangent of the angle opposite the other is to the tangent of 
the other. 

Let ABC be a triangle, right- 
angled at B, on the surface of a 
sphere whose centre is O ; then 

R : sin AB ; : tan BAC : tan BC. 

DrawBF perpendicular to AO ; 
from F draw FD, in the plane 
ACO, perpendicular to AO, and 
meeting OC produced in D ; and 
join BD. 




170 



BOOK V. 



Because AO is perpendicular 
to FB and FD, it is perpendicular 
to the plane FBD (Geoni. 322) ; 
and therefore the plane ABO, 
passing through the straight line 
AO, is perpendicular to FBD, 
(Geom. 328). But the plane BOO 
or OBD is also perpendicular to 
ABO, since the angle ABC is a 
right angle ; hence BD is perpen- 
dicular to the plane ABO (Geom. 
329), and is therefore perpendicular to FB and BO. 

We have therefore BD for the tangent of BC, BF for the 
sine of AB, and the angle BFD measures the spherical angle 
BAC (Geom. 432 and 336). 

The right-angled plane triangle FBD gives 




R : FB ! : tan BFD : BD (46) ; 

tan BAC : tan BC. 



hence, B \ sin AB 



Therefore, in every right-angled spherical triangle, &c. 

(161.) Cor. — In every right-angled spherical triangle, 
Badius is to the tangent of either of the sides containing the 
right angle, as the cotangent of the angle opposite to that 
side is to the sine of the other side about the right angle. 

For the triangle FBD gives 

R I BD : : tan BDF, or cot BFD : BF (46) ; 
that is, B : tan BC : : cot BAC : sin AB. 



THEOREM IV. 



(162.) In every right-angled spherical triangle, Radius is 
to the cotangent of either of the angles adjacent to the hypo- 
thenuse, as the cotangent of the other is to the cosine of the 
hypothenuse. 




SPHERICAL TRIGONOMETRY. 171 

Let the spherical triangle, 
ABC be right-angled at B ; then 

r : cot A : : cot acb : cos AC. 

Produce each side of this tri- 
angle until it meets the arc DF 
of a great circle described from 
A as its pole. ' 

The great circle ABD, passing through the pole A of the 
great circle DGF, is a secondary to the latter circle, and 
the angles ADF and AGF are right angles (Geom. 442); 
also, ABF and ADF being right angles, the arcs BF and DF 
meet in the pole F of the arc AD. 

The arcs ABD, ACG and BCF are therefore quadrants 
(Geom. 440) ; so that CG is the complement of AC, and GF 
ofDG. 

In the right-angled triangle CGF, we have 

R : tan GF : : cot GCF : sin CG (161). 

Hence, regarding tan GF as the cotangent of DG which 
measures the angle A, and sin CG as the cosine of AC, — and 
considering that the vertical angles at C are equal — we have 
in the triangle ABC, 

r : cot A : : cot acb : cos AC. 

Therefore, in every right-angled spherical triangle, &c. 

(163.) Cor. — ■ In every right-angled spherical triangle, 
Radius is to the cotangent of thedrypothenuse, as the tangent 
of either of the other two sides is to the cosine of the angle 
opposite to the third side. 

For in the right-angled triangle CGF, we have 

R : tan CG : : cot F : sin GF (161). 

But CG is the complement of AC, the cotangent of F, or 
of its measuring arc BD, is the tangent of AB, and the sine 
of GF is the cosine of DG which measures the angle A ; so 
that 

R : cot AC ; : tan AB : cos A. 



172 BOOK V. 



THEOREM V. 



(l 64.) In every spherical triangle, the sines of the angles 

are proportional to the sines of the opposite sides. 

Let ABC be any spherical tri- 
angle ; then 




sin A : sin B ! I sin BC : sin AC. 

Let CD be an arc of a great 
circle at right angles with AB 
or AB produced. Then ADC and BDC are right-angled 
spherical triangles ; so that 

R : sin AC : ! sin A : sin CD, 
and K : sin BC \ \ sin B : sin CD (157) ; 

which give sin AC . sin A=R . sin CD, 
and sin BC . sin B=R . sin CD. 

Hence, sin AC . sin A=sin BC . sin B ; 

or, sin A : sin B ; \ sin BC : sin AC (Geom. 180). 

Therefore, in every spherical triangle, the sines of the 
angles, &c. 

THEOREM VI. 

(165.) In every spherical triangle, the tangents of any two 
of the angles are reciprocally proportional to the sines of the 
segments of the included side, made by an arc of a great circle 
drawn perpendicular to that side, from the opposite vertex. 

In the right-angled triangles 
ADC and BDC, 

R I sin AD : ! tan A : tan CD, 

(160.) 
R : sin BD : : tan B : tan CD ; 

which give sin AD . tan A=R . tan CD, 
and sin BD . tan B=R . tan CD. 




SPHERICAL TRIGONOMETRY. 173 

Hence, in the spherical triangle ABC, 

tan A : tan B : : sin BD : sin AD (Geom. 180). 
Therefore, in every spherical triangle, the tangents, &c. 

THEOREM VTI. 

(166.) In every spherical triangle, the cosines of any two 
are proportional to the cosines of the segments of the 
third side, made by an arc of a great circle, drawn perpen- 
dicular to that side from the opposite vertex. 

In the right-angled triangles ADC and BDC, in the pre- 
ceding figure, 

E : cos AD : cos CD : cos AC, 
and R : cos BD : cos CD : cos BC (158). 

Hence, in the spherical triangle ABC, 

cos AD : cos BD \ \ cos AC : cos BC (Georn. 191). 
Therefore, in every spherical triangle, the cosines, &c. 

THEOREM VIII. 

(16T.) In every spherical triangle, the cotangents of any 
two sides are proportional to the cosines of the segments of 
the included angle, made ~by an arc of a great circle drawn 
from the vertex of that angle perpendicular to the opposite 
side. 

In the right-angled triangles ADC and BDC, in the pre- 
ceding figure, 

R I cot AC : : tan CD : cos ACD, 
and E : cot BC ; : tan CD : cos BCD (163). 

Hence, in the spherical triangle ABC, 

cot AC : cot BC : : cos ACD : cos BCD (Geom. 191). 
Therefore, in every spherical triangle, the cotangents, &c. 



174 BOOK v. 




THEOREM IX. 

(168.) In every spherical triangle, the cosine of any side is 
equal to the product of the cosines of the other two sides, plus 
the product of the sines of those two sides into the cosine of 
their included angle, when the radius is unity. 

Let ABC be a triangle on 
the surface of a sphere whose 
centre is O. Let AD be the 
tangent, and OD the secant 
of the arc AB ; AF the tan- 
gent, and OF the secant of 
the arc AC. 

Let a represent the side BC opposite the angle A, b the 
side AC opposite the angle B, and c the side AB opposite the 
angle C, in the given triangle ABC. Then 

AD=tan c ; OD=sec c ; AF=tan b ; OF=sec b. 

Considering that the arc a is the measure of the angle DOF 
(G. 260), and that the angle DAF is equal to the spherical A 
in the triangle ABO (G. 448), we have, from the triangles 
ODF and ADF,— radius being unity, 

DF 2 =OD 2 +OF 2 -20D . OF cos a, 
DF 2 =AD 2 +AF 2 -2AD . AF cos A (50). 

By subtracting the second equation from the first, — 
observing that OD 2 -AD 2 and OF 2 -AF 2 are each equal to 
AO 2 (Geom. 149), equal to unity 2 =l, — we have, after dividing 

0=1— sec c sec b cos #+tan c tan b cos A. 

From Article (26. ..1 and 2) we have 

1 1 sin c 

sec c = ; sec o = ? ; tan c = ? 

cos c ' cos b > cos o 

z sin b 

tan b = -. 

cos b 

By substituting these values in the last equation — multi- 



SPHEEICAL TRIGONOMETRY. 175 

plying the resulting equation by cos c cos b — and then 
dividing by 2, we shall find 

cos a=cos b cos <?+sin b sin c cos A. 

Therefore, in every spherical triangle, the cosine of any 
side, &c. 

THEOREM X. 

(169.) In every spherical triangle, the cosine of any angle 
is equal to the product of the sines of the other two angles into 
the cosine of their included side, minus the product of the 
cosines of the same two angles, when the radius is unity. 

Let ABC and A'B'C' be any two spherical triangles on the 
same sphere which slyq polar to each other (Geom. 452). Let 
a represent the side opposite to the angle A, a' the side 
opposite to the angle A 7 , &c, in each triangle ; then 

A=180 o -a', which gives a'=180°-A 
B=180°-&' 5 which gives J'=180°-B 
O=180°-c / , which gives c'=180°-C 
also A'=180°-a (Geom. 453). 

By substituting these values of a', b', and c' in the equation 

cos <z /= cos V cos c'+sin V sin c' cos A' (168) ; 

observing that cos (180°— A) is equal to —cos A, &c, and sin 
(180-B) equal to sin B, &c. (20), we shall find 

—cos A=— cos Bx— cos C-f-sin B sin Cx— cos a. 

When the signs of all the terms are changed (Alg. 117), we 

have 

cos A=sin B sin G cos a— cos B cos C. 

In like manner we might obtain, 

cos B=sin A sin C cos b— cos A cos C ; 
cos C-=sin A sin B cos c— cos A cos B. 

Therefore, in every spherical triangle, the cosine of any 
angle, &c. 



176 BOOK V. 



THEOREM XI. 



(170.) If &, b, and c represent the sides which are respect- 
ively opposite to the angles A, B, and C of any spherical 
triangle, and S represent half the sum of the sides, radius 
being unity / 



/sin (S-&) sin (S-e) 
sin \ A.—\J — 



sin o sin o 



By substituting 1—2 sin 2 £ A for cos A (31.. .14) in the 
formula 

cos a=cos b cos c+sin b sin c cos A (168), 

and considering that cos b cos c+sin b sin c=cos (b—c) 
(31...4), we find 

cos #=cos (b—c)— 2 sin 5 sin c sin 2 \ A, 
or 2 sin b sin c sin 2 ^ A=cos (b— c)— cos #. 

The second member of the last equation is equal to 

2 sin | (a+b-c) sin J (a+c-J), 31...12). 

By substituting this equivalent, dividing both sides by 2 
sin 5 sin c, and considering that, as #+&+c=2S, a+5— <? is 
equal to 2S— 2c, and #+c— b is equal to 2S— 25, we shall 
obtain, after extracting the square root, 

/sin (S-&) sin (S-c) 

sin ^ A= V -7-7"^ 

sm b sin c 

Therefore, if &, 6, and c represent the sides, &c. 
(171.) #w\ — In like manner we should find 



/sii 

*b=v- 



/sin (S—a) sin (S—c) 
sm 

sm a sm c 



. . _. /sin (S— a) sin (S— b) 
sin f C=y — 



sin a sin & 



SPHERICAL TRIGONOMETRY. 177 

SOLUTION OF SPHERICAL TRIANGLES. 

(172.) The solution of a Spherical Triangle consists in 
computing any three of its six parts when the other three are 
given. 

A Spherical Triangle diners from a plane triangle in this 
respect, that any three parts of the former determine the other 
three, whereas in a plane triangle the three angles do not 
determine the sides. 

I. Right-angled Spherical Triangles. 

(173.) The Theorems which have been demonstrated for 
right-angled spherical triangles, with their corollaries, furnish 
proportions for the solution of every case that can occur 
among such triangles. 

These several proportions have all been embraced, by Lord 
Napeer, in one general Theorem, designed to relieve the 
memory. 

In this Theorem the parts of the triangle considered, five 
in number, are the two sides containing the right angle, the 
complements of the other two angles, and the complement of 
the hypothenuse. Any one of these being called the middle 
part, the two which lie next to it, on different sides, are 
called the adjacent parts, and the other two the opposite 
parts. 



spherical triangles. 

(174.) Radius multiplied into the sine of the middle part 
is equal to the product of the tangents of the adjacent parts, 
or to the product of the cosines of the opposite parts. 

Let the spherical triangle ABC be 
right-angled at B. Denote the sides 
by the small letters a, b, c, cor- 
responding to the capitals at the 
vertices opposite to the respective 
sides; then the five parts embraced 





178 BOOK V. 

by the Theorem are the sides a and 
c containing the right angle B, the 
complements of the angles A and C, 
and the complement of the hypo- 
thennse b. 



If we take the side a for the middle part, the side c and 
the complement of the angle C will be the adjacent parts, the 
complement of the angle A and the complement of the side 
b will be the opposite parts. 

R : sin b \ \ sin A : sin a (157) ; 

R I tan c \ '. cot C : sin a (161) ; 

hence, R . sin &=tan c . cot C=sin b, sin A. 

The cot C is the tangent of the complement of 0, the sin b 
and sin A are the cosines of the complements of b and A ; 
therefore, Radius into the sine of the middle part, &c. 

If we take the complement of the side b for the middle 
part, the complements of the angles A and C will be the 
adjacent parts, the sides a and c will be the opposite parts. 

R I cos c \ ! cos a : cos b (158) ; 

R : cot J. : : cot C : cos b (162) ; 

hence, R . cos &=cot A . cot C=cos a . cos c. 

The cos b is the sine of the complement of b, the cot A and 
cot C are the tangents of the complements of A and C ; there- 
fore, Radius into the sine of the middle part, &c. 

If we take the complement of the angle A for the middle 
part, the side c and the complement of the side b will be the 
adjacent parts, the side a and the complement of the angle 
will be the opposite parts. 

R : cos a \ \ sin C \ cos A (159) ; 

R : cot b ! I tan c : cos A (163) ; 

hence, R . cos A=cot b . tan c=cos a . sin C. 



SPHERICAL TRIGONOMETRY. 179 

The cos A is the sine of the complement of A, the cot b is 
the tangent of the complement of b, the sin is the cosine of 
the complement of C ; therefore, Radius into the sine of the 
middle part, &c. 

In the preceding Equations we have taken, successively, 
for the middle part, one of the sides containing the right 
angle B, the complement of the hypothenuse b, and the com- 
plement of one of the angles adjacent to the hypothenuse. 
Those equations therefore contain all the varieties of arrange- 
ment of 1$ after' 8 Jive parts, proposed by his Theorem. 



Inferences from Preceding Equations. 

(175.) In a right-angled spherical triangle, neither of the 
sides containing the right angle, and the opposite angle, can 
be the one less and the other greater than 90°. 

This follows from the equation 

R . cos A==cos a . sin C, found above. 

For the radius R is positive (25), and sin C is positive, since 
the angle C is less than 180° (19) ; cos A and cos a must 
therefore be both positive, or both negative, in order that the 
two sides of the equation may have the same algebraic sign 
(Alg. 42). The sides and angles of a spherical triangle being 
always less than 180°, it follows that the angle A and the 
side a will be in the same quadrant (19). 

(176.) When the sides containing the right angle are both 
less, or both greater, than 90°, the hypothenuse is less than 
90° ; when one is less and the other greater, the hypothenuse 
is greater than 90°. 

This follows from the equation 

R . cos 5= cos a . cos c, found above. 

For R being always positive, cos b will be positive, or 
negative, according as cos a and cos c have the same, or con- 
trary, signs ; the hypothenuse b will therefore be in the first 



180 BOOK V. 

quadrant, or in the second, according as a and c are both less 
or greater, or one less and the other greater, than 90° (19). 

It also follows that when the sides containing the right 
angle are each 90°, the hypothenuse is 90°. 

Any two parts of a Spherical Triangle are said to be of 
the same species, or of the same affection, when they are both 
greater, or both less, than 90°. 



GENERAL RULE 

For the Solution of Eight-angled Spherical Triangles. 

1. Of the two given parts (exclusive of the right angle) and 
the required part, take that one for the middle part (in the 
nomenclature of Napier) which will cause the other two parts 
to be both adjacent or both opposite (173). 

2. Form an Equation according to Napier's Theorem (174) ; 
convert the equation into a proportion (Geom. 180), with the 
required part for the last term, and compute that term as a 
fourth proportional. 

EXAMPLE I. 

In a right-angled spherical triangle ABC, the angle A is 
80° 40', and the hypothenuse o is 105° 34'. Find the remain- 
ing parts of the triangle. 

c 

1. To find the side a. 

The two given parts are the angle A 
and the side b. If the side a be taken 
for the middle part, the complements 
of A and b will evidently be the oppo- 
site parts', and we shall have -^B 

R sin a=cos (comp. A) cos (comp. b)= sin A sin b (174). 

This equation, converted into a proportion, with the 
required term sin a for the fourth term, becomes 




SPHERICAL TRIGONOMETRY. 181 

Eadius 10 10 comp. log. 0.000000 

is to sin A 80° 40 / 9.994212 

as sin b 105° 34' 9.983770 

is to sine of a 71° 54' 33" 9.977982 

The side a is tlms found from its sine. Every sine cor- 
responds to two arcs which are supplements of each other 
(20) ; but in this triangle the side a is less than 90°, because 
the angle A is less than 90° (175). 

2. To find the side c. 

If the complement of the angle A be taken for the middle 
part, the side c and the complement of b will be adjacent 
parts / and we shall have 

R cos A=tan c cot b (174) ; 

the sine of the complement of A being the cos A, and the 
tangent of the complement of b the cot b. 

When this equation is converted into a proportion, with 
the required term tan c for the fourth term, it becomes 

cot b 105° 34 / comp. log. 0.555053 

is to R 10 10 10.000000 

as cos A 80° 40' 9.209992 

is to tan of c 149° 47' 37" ...... 9.765045 

The side c is thus found from its tangent. Irrespective of 
its sign, the tangent does not show whether the side c is less 
or greater than 90° (20...2). We must therefore determine 
from the equation employed whether tan c is positive or 
negative. 

The first side of the equation is positive, since R and cos A 
80° 40' are both positive (Alg. 42} ; the other side is there- 
fore positive, and cot b 105° 34' being negative (19), it follows 
that tan c is also negative. The side c is therefore greater 
than 90°, being the supplement of the degrees corresponding 
to the logarithmic tangent 9.765045 in the Table. 

3. To find the angle C. 

Take the complement of the hypothenuse b for the middle 



182 



BOOK V. 



part ; the complements of the angles A and C will be the 
adjacent parts / and hence 

E cos 5=cot A cot C (174). 

The angle C will be found from its cotangent. ~By reason- 
ing in the same manner as for tan c under the preceding 
Example, it will be found that cot C is negative, and the 
angle C therefore greater than 90° (19). 



EXAMPLE II. 




In the spherical triangle ABC, right-angled at B, the side 
c is 29° 12', and the angle C 37° 26'. Find the side a. 



Take the required side a for the middle 
pari; the given side c and the comple- 
ment of the given angle C will then be 
the adjacent parts, and we shall have 

R sin #=tan c . cot C (174). 

By converting this equation into a pro- 
portion, — making the required term sin a 
the fourth term. 



R 10 10 comp. log. 0.000000 

is to tan c 29° 12' 9.747319 

as cot C 37° 26' 10.116066 

is to sine of a 46° 54' or 133° 6' . . . 9.863385 

The side a is thus found from its sine. Every sine cor- 
responds to two arcs which are supplements of each other, 
and in this case it is evident that there are two triangles ABC 
and ABC, both right-angled at B, containing the given side 
<?, equal given angles C and C, and having the remaining 
parts of the one supplementary to the remaining parts of the 
other ; — all which is plain from conceiving the sides CA and 
CB to be produced until they meet in C diametrically oppo- 
site to C (Geom. 434...5). 



SPHERICAL TRIGONOMETRY. 183 

From this it may be concluded, in general, that 

(177.) The solution of a right-angled spherical triangle is 
ambiguous whenever one of the given parts is one of the sides 
containing the right angle, and another the angle opposite 
to that side. 

1. In the spherical triangle ABC, right-angled at B, the 
angle A is 23° 28', and the side c 49° 17'. Find the hypo- 
thenuse b. Ans. b 51° 43'. 

2. In the spherical triangle ABC, right-angled at B, the 
hypothenuse b is 66° 32', and the side a 37° 48'. Find the 
angle C. Ans. C 70° 19'. 

3. In the spherical triangle ABC, right-angled at B, the 
side a is 59° 38', and the side e 48° 24'. Find the angles A 
and C. Ans, A 66° 20' ; C 52° 33'. 

4. In the spherical triangle ABC, right-angled at B, the 
angle A is 31° 51', and the hypothenuse b 113° 55'. Find 
the sides a and c. Ans. a 28° 50' ; c 117° 34'. 

5. In the spherical triangle ABC, right-angled at B, the 
angle A is 37° 25', and the side a 36° 31'. Find all the 
remaining parts of the triangle. 

Ans. b 78° 20', or 101° 40' ; c 75° 26', or 104° 34' ; 
C81° 12', or 98° 48'. 



H. Solutions of Qijadrantal Triangles. 

(178.) A Quadrantal Spherical Triangle is one which has 
a side equal to a quadrant, or 90°. Such a triangle, with 
two other parts also given, may be solved through the 
medium of its polar triangle (Geom. 452), in which the angle 
opposite to the quadrantal side of the former is a right angle. 

6. In the spherical triangle ABC the side b is a quadrant, 
or 90°, the side c 48°, and the angle C 42° 12'. Find the 
side a. 9 



184; BOOK V. 

« 

Let A'B'C be the polar triangle; then 

B'=180°-90°:=90 o ; 
C'=180 o -48 o =132° ; 
C=180°-c', or c'=180°-(42° 12')=137° 48' (Geom. 453). 

In the triangle A'B'C, right-angled at B', we have there- 
fore the angle C, and the side c', to find the angle A'. 

A'=180-a, or a=180-A'. 

The angle A' will be ambiguous, and therefore the required 
side a will also be ambiguous (177). 

Ans. a 64° 35' or 115° 25'. 

7. In the spherical triangle ABC the side b is a quadranfy 
or 90°, the angle A 54° 43', and the angle C 42° 12'. Find 
the angle B. Ans. B 115° 20'. 

8. In the spherical triangle ABC the side b is a quadrant, 
or 90°, the side c 115° 9', and the angle A 115° 55' Find 
the angle C. Ans. C 117° 34'. 



HI. Solutions of Oblique-angled Spheeical Triangles. 

case I. 

(179.) When a Side and the opposite Angle are two of the 
given parts. 

The sine of any side \ the sine of the opposite angle : th# 
sine of either of the other sides : the sine of the opposite 
angle. 

The sine of any angle \ the sine of the opposite side : the 
sine of either of the other angles : the sine of the opposite 
side (164). 

After the application of one or the other of these propor- 
tions, the triangle will have two sides and the two angles 
opposite to those sides given. The remaining parts may 
then be computed from two right-angled triangles formed by 
drawing an $rc of a great circle from the common extremity 



SPHERICAL TRIGONOMETRY. 



185 



of the two given sides perpendicular to the third side, or to 
that side produced. 



EXAMPLE. 



In the spherical triangle ABC the side AC is 85°, BC 70°, 
and the angle A 60°. Find the other parts of the triangle. 



1. First find the angle ABC. 




Sin BC 70° : sin A 60° ; : sin 

AC 85° : the sine of ABC 

(164). 



The angle ABC will thus be 
found from its sine. Every sine corresponds to two arcs, or 
angles, which are supplements of each other ; and in the pre- 
sent case there are two triangles, ABC and AB'C, containing 
the given sides AC and BC or B'C, and the given angle A. 

The angles ABO and AB'C are supplementary to each other, 
since the latter is equal to CBB' (Geom. 463), and the two 
angles at B are together equal to two right-angles. 

Draw CD, an arc of a great circle, perpendicular to ABB'. 

The given angle A being acute, it is plain that DC is less 
than a quadrant (Geom. 442 and 440), and that the angle 
DBC is also acute : hence ABC is obtuse. 

If the angle A were obtuse, DC would be greater than a 
quadrant, the angle DBC also obtuse, and ABC acute. 

2. To find the angle ACB. 
In the right-angled triangle ADC the side AC and the 

angle A are given ; find the angle ACD. In the right-angled 
triangle BDC the side BC is given, and the angle DBC is the 
supplement of ABC ; find the angle BCD. Then the angle 
ACD-BCD gives the angle ACB. 

3. To find the side AB. 

Sin A : BC ; : sin ACB ■ sin AB. 
The sine of AB corresponds to two arcs which are supple- 



186 BOOK Y. 

merits of each other ; but the proper one to be taken will be 
known from a comparison of the angles A and ACB ; the 
greater of two angles having a greater side opposite to it 
(Geom. ±66). 

The angle ACB' would be found by taking the sum of the 
angles ACD and DCB', in the right-angled triangles ADC 
and B'DC. The side AB' would then be found in the same 
manner with AB. 



When a Side and the opposite Angle are two of the given 
parts of a Spherical Triangle, there may be only one triangle, 
or there may be two, according to the following principles. 

(180.) When the sine of a given side opposite to a required 
angle is less than the sine of another given side, there is hut 
one triangle — when greater, there are two triangles, fulfilling 
the given conditions. 

This may be shown as follows ; from the formula of 
Theorem IX. (168), 

. cos a— cos b cos c 

COS A= — — ; 

sin o sin c 

Let a be the given side opposite to the required angle A, 
and b another given side ; and let sin a be less than sin b. 

The cos a is greater than cos b ; and also greater than cos 
b cos c, since cos c is less than radius 1. The numerator in 
the equation will therefore be positive, or negative, according 
as cos a is positive, or negative. 

The denominator is positive, since all sines in the first two 
quadrants are positive (19). The sign of the second member 
will therefore be the same as that of cos a (Alg. 49) ; and 
since the two members must have the same sign, it follows 
that the required angle A will be greater, or less, than 90°, 
according as the given side a is greater, or less, than 90° ; so 
that A will have but one value. 

Secondly ; let sin a be greater than sin b. The cos a will 



SPHERICAL TRIGONOMETRY. 187 

then be less than cos h ; and the side c may be of such value 
that the cos a will be less than cos o cos o. The sign of the 
second member of the equation will thus depend on that of 
cos c ; that is, o may be of such value as to make cos A 
either positive or negative, and A either less or greater than 
90°. 

This last case is illustrated in the preceding Example, in 
which the sine of the side AC is greater than that of BC. 

(181.) When the sine of a given angle opposite to a required 
side is less than the sine of another given angle, there is hut 
one triangle — when greater, there are two triangles, fulfilling 
the given conditions. 

This may be shown from the equation 

cos A+cos B cos C ^ -^- 

cos a= : , (169), 

sin B sin C 

by a course of reasoning analogous to that which has been 
pursued under the preceding proposition. 



9. In the spherical triangle ABC the side AC is 57° 30', 
the side BC 115° 20', and the angle A 126° 37'. Find the 
side AB. Ans. AB 82° 26'. 

10. In the spherical triangle ABC the angle A is 51° 30', 
the angle B 59° 16', and the side BC 63° 50'. Find the side 
AC. Ans. AC 80° 19', or 99° 41'. 

11. In the spherical triangle ABC the side AB is 114° 30', 
the side BC 56° 40', and the angle 0-125° 20'. Find the 
remaining angles. Ans. A 48° 30', and B 62° 54'. 

12. In the spherical triangle ABC the angle A is 50° 12', 
the angle B 58° 8', and the side BC 62° 42'. Find the 
remaining parts of the triangle. 

Ans. The side AC 79° 12' 10", or 100° 47' 50" ; the 
angle C 130° 54' 30", or 156° 15' 6" ; the side 
AB 119° 3' 29", or 152° 14' 18". 



188 BOOK V. 

CASE II. 
(182.) When two Sides and the included Angle are given. 

From the extremity of one of the two given sides draw an 
arc of a great circle perpendicular to the other given side, or 
to that side produced. 

There will thus be formed a right-angled triangle contain- 
ing the given angle and one of the given sides, from which 
the segments of the other given side may be computed, and 
thence the remaining parts of the given triangle. 

EXAMPLE. 

In the spherical triangle ABC the side AB is 120° 47', the 
side AC 80° 19', and the included angle A 51° 30'. Find 
the other parts of the triangle. 

Draw CD, an arc of a great 
circle, perpendicular to AB 
Then in the right-angled tri- 
angle ADC, we have the side 
AC and the angle A ; find 
AD. (173). 

Having found the segment AD, the segment DB will be 
found by subtracting AD from AB. 

To find the angle B. By Theorem YT. (165), we have 

sin DB : sin AD ; ; tan A : tan B. 

To find the side BC. By Theorem Y1L (166), we have 

cos AD : cos DB ; ; cos AC : cos BC. 

To find the angle ACB. 

Sin AC : sin B : : sin AB : sin ACB (164). 

The sine of ACB corresponds to two angles which are sup- 
plements of each other ; but the proper one to be taken will 
be known from a comparison of the side AB with AC, or 




SPHERICAL TRIGONOMETRY. 189 

BC ; the greater of two sides having a greater angle opposite 
to it (Geom. 466). Ans. B 59° 16'; BO 63° 50'. 

13. In the spherical triangle ABC, the side AB is 76° 20', 
the side BC 119° 17', and the included angle B 52° 5'. Find 
the side AC. Ans. AC 66° 5' 38". 

14. In the spherical triangle ABC, the side AC is 57° 30', 

the side AB 82° 27', and the included angle A 126° 37'. 
Find the remaining parts of the triangle. 

Ans. B 48° 30' ; C 61° 42'; BC 115° 20'. 



CASE III. 

(183.) When two Angles and the included Side are given. 

. From the vertex of one of the two given angles draw an 
arc of a great circle perpendicular to the opposite side, or to 
that side produced. 

There will thus be formed a right-angled triangle con- 
taining one of the given angles and the given side, from 
which the segments of the divided angle may be computed, 
and thence the remaining parts of the given triangle. 

EXAMPLE. 

In the spherical triangle ABC the angle A is 51° 30', the 
side AC 80° 19', and the angle C 131° 30. Find the other 
parts of the triangle. 

Draw CD, an arc of a great 
circle, perpendicular to AB. 
Then in the right-angled tri- 
angle ADC, we have the side 
AC, and the angle A ; find 
the angle ACD (173). 

Having found the angle ACD, the angle DCB will be 
found by subtracting ACD from the given angle C. 

To find the side BC. By Theorem VIII. (167), we have 

cos ACD : cos BCD ; : cot AC : cot BC. 




190 BOOK V. 

Having found the side BC, the angle B and the side AB 
may be determined from the proportion between the sides 
and those of the opposite angles (164). Ans. BC 63° 50'. 

15. In the spherical triangle ABC, the angle A is 48° 30', 
the side AC 83° 12', and the angle C 125° 20'. Find the 
side BC. Ans. BC 56° 39'. 

16. In the spherical triangle ABC, the angle A is 126° 37', 
the side AB 57° 30', and the angle ABC 61° 41'. Find the 
remaining parts of the triangle. 

Ans. C 48° 30'; BC 115° 20'; AC 82° 26'. 



CASE IY. 
(184.) When the three Sides are given. 
The angles may be found from the sides by the formula 



. i . ^ /sin(S-5)sm( S-.)^ AN> 

Bin i A= V — — (170) ; 

sin o sin c 

in which A will be any required angle, S half the sum of the 
three sides, b and c the sides containing the required angle. 

This formula is true only when the radius is unity ; but it 
may be adapted to any other radius B (27). Squaring both 
sides, and introducing the factor B 2 into the second member, 
so as to make all the terms homogeneous when the formula 
is cleared of its denominator, we obtain 

B 2 sin (S-5) sin (S-c) 

sin 2 1 A- . z . ; 

sin o sin c 

in which the radius may be 10 10 



EXAMPLE. 

In the spherical triangle ABC the side AB is 120° 47', BC 
63° 50', and AC 80° 19'. Find the angles A, B, and C. 




SPHERICAL TRIGONOMETRY. 191 

Denoting the sides which c 

are respectively opposite to 
the angles A, B, and C by 
the corresponding small let- 
ters, we have 

c 

a=63° 50', 3=80° 19' c 120° 47'. 
To find the angle A. The value of S in the formula is 
i of (63° 50'+80° 19'+120° 47')=132° 28' ; 
S-&=52° 9 7 , and S-c=ll° 41'. 

The logarithmic calculation — using the complements of the 
logarithms of the divisors sin h and sin c (9), is as follows : 

E 2 (Alg. 310) 20.000000 

sin (S-J) 52° 9' 9.897418 

sin (S-c) 11° 41' 9.306430 

sin b 80° 19'. ...... comp. log. 0.006232 

sin c 120° 47' comp. log. 0.065952 

Logarithm of sin 2 \ A=19.276032 

We reject 20 from the sum of the logarithmic numbers 
because two complements are used. The result is the loga- 
rithm of the square of the sine of half the angle A ; 

then \ of 19.276032=9.638016 is the log. sin of \ A (S...4). 

This sine, in Table II., corresponds to 25° 45' 18" ; then 

the angle A is (25° 45' 18 // )X2=51° 30' 36". 

By this method we find half the required angle from its 
sine. This half angle is necessarily acute, since the whole 
angle is less than 180°. No ambiguity, therefore, attaches 
to an angle thus found, as is sometimes the case with an 
angle found from its sine. 

The angles B and G might be found from similar formulas ; 
but, the angle A having .been determined, the others may be 

9* 



192 BOOK V. 

more readily obtained from the proportion between the sines 
of the sides and those of the opposite angles (164). 

Ans. B=59° 17'; 0=131° 29'. 

17. In the spherical triangle ABC the side AB is 82° 28', 
BC 115° 20', and AC 57° 30'. Find the angle A. 

Ans. A 126° 34'. 

18. In the spherical triangle ABC the side AB is 59° 12', 
BC 81° 17', and AC 114° 3'. Find the angles A, B, and C. 

Ans. A 62° 39' 42" ; B 124° 50' 52" ; C 50° 31' 42". 

CASE V. 

(l§5.) When the three Angles are given. 

The sides of a spherical triangle may be found from its 
angles through the medium of the polar triangle (Geom. 452) ; 
as shown by the following 

EXAMPLE. 

In the spherical triangle ABC the angle A is 50°, the angle 
B 60° and the angle C 140°. Find the sides AB, BC, and 
AC. 

Let A'B'C be the polar triangle / and denote the sides 
opposite to the angles A, A', &c, in the two triangles, by 
the corresponding small letters a, a\ &c. Then the angle 



>o , 



A, 50°=180 o -a', or «'=130 c 

B, 60°=180°-Z/, or ^=120°; 

C, 140°=180°-c', or c'= 40° (Geom. 453). 

From the sides a\ b' \ & of the polar triangle find its angles 
A', B', a, as in Case IY. Then 

A'=18O°-0, which gives a=180°-A'; 
B'=180°--&, which gives 5=180°-B' ; 
C'=lS0 o -~c, which gives c=180°~C'. 



SPHERICAL TRIGONOMETRY. 193 

Tims will be determined the sides a, 5, and c of the given 
triangle. Ana. a 62° 10' 57" ; h 89° 5' 33" ; c 122° 5' 12". 

19. In the spherical triangle ABC, the angle A is 51° 30', 
the angle B 59° 16', and the angle C 131° 30'. Find the side 
AC. Ans. AC 80° 19' 14". 

20. In the spherical triangle ABC, the angle A is 36° 8', 
the angle B 46° 19', and the angle C 104.° Find the sides 
AB, BC, and AC. 

Ans. AB 42° 9' ; BC 24° 4' ; and AC 30°. 



APPLICATIONS OF SPHERICAL TRIGONOMETRY. 

(186.) The most important Applications of Spherical Tri- 
gonometry are the solutions of spherical triangles formed by 
arcs of great circles of the Earth, or of the Celestial Sphere, 
as required for certain purposes in Geography, Navigation, 
and Astronomy. 

Among these applications is the determination of latitudes 
and longitudes, as in Nautical Astronomy (155) ; but it does 
not come within the scope of the present work to describe 
the observations and the computations requisite for such 
purposes — which belongs properly to a treatise on the general 
subject of Practical Astronomy. 

We shall therefore only present a few general Problems 
referring to the terrestrial and celestial spheres, in which, as 
in the Problems of Book IY., the given latitudes and longi- 
tudes are supposed to have been determined astronomically. 



194 



BOOK V. 



(187.) Shortest Distance between two Places whose Latitudes 
and Longitudes are given. 

Let A and B be the two places ; let P be the nearest pole 
of the earth, PE and PQ meridians passing through the 
points A and B, and EQ an arc of the equator. 

The shortest line that can be drawn, 
on the level surface of the earth, be- 
tween the points A and B, is the arc 
AB of a great circle (Geom. 437). 

The latitude AE of the point A 
subtracted from PE, 90°, will leave 
the co-latitude AP ; and the latitude 
BQ of the point B subtracted from 
PQ, 90°, will leave the co-latitude 
BP. Also the difference of longitude, 
EQ, of the points A and B is the 
measure of the angle APB (Geom. 
447). 

In the spherical triangle ABP, there will therefore be given 
the sides AP and PB, and the included angle APB, from 
which the side AB may be computed (182). 

AB will be found in degrees of a great circle of the earth. 
These degrees may be reduced to nautical miles by allowing 
60 miles to a degree ; or to statute miles by allowing 69.154 
miles to a degree — this being regarded as the average length 
of a degree on the earth. 




(l§§.) The Course of the shortest Distance on the Earth 
between tvio Places vjhose Latitudes and Longitudes are 
given. 

Let A and B, in the preceding diagram, be the two places; 
the different lines in that diagram representing the same 
elements as in the preceding problem. 

The angle PAB, which may be computed from the sides 
AP and PB and the included angle APB (182), is the course 
of the arc AB at the point A ; but since the meridians are 



SPHERICAL TRIGONOMETRY. 195 

not parallel, any great circle, except the equator, will cut 
any two meridians at unequal angles, and the course of such 
circle will therefore continually vary with respect to the 
successive meridians which it crosses. 

Thus Pe> being a meridian intersected by the arc AB in the 
point s, the angle PsB is the course of the arc AB at the 
point s. This angle may be computed in the triangle PsB, 
after the latitude and longitude of the point s have been 
determined. 

A Steamship, in ordinary weather, pursues the course of 
a great circle, from one port to another. The course is recti- 
fied daily, from the latitude and longitude the ship is in, or 
for about every five degrees difference of longitude that she 
makes. Her track is thus made to coincide, nearly, with an 
arc of a great circle. 

21. Find the distance, on a great circle of the earth, from 
Philadelphia, in latitude 39° 57' K, longitude 75° 10' W,, to 
St. Louis, in lat. 38° 37' K, longitude" 90° 15' W. 

Ans. 811.4 miles. 

22. Find the distance, on a great circle of the earth, from 
Savannah, in latitude 32° 4' K, longitude 80° 58' W.., to 
Lizard Point, in latitude 19° 58' 1ST., longitude 5° 11' W. ; also 
the course on which a ship must start from the former place, 
to reach the latter on the arc of a great circle. 

Ans. 3437 miles ; K 47° 49' E. 

ASTRONOMICAL PROBLEMS. 

(l§9.) " The solution of Astronomical Problems forms one 
of the most useful and agreeable applications of the theory 
of Spherical Trigonometry. To such inquiries the theory 
itself, no doubt, owes its origin, as well as many of the suc- 
cessive improvements which it has gradually received ; so 
that a specimen of its use in the solution of astronomical 
problems may reasonably be looked for in a book on Trigo- 
nometry." 

The triaDgles involved in these problems are formed by 
arcs of great circles of the 



196 BOOK V. 



CELESTIAL SPHERE. 



(190.) The Celestial or starry Sphere is the spherical con- 
cave surrounding the earth, and concentric with it, in which 
all the heavenly bodies appear to be situated. 

The revolution of the earth on its axis, from west to east, 
once every 24 hours, causes an apparent revolution of the 
celestial sphere, from east to west, in the same time, around 
the celestial axis, which is the earth's axis produced to the 
starry concave. 

The celestial poles, north and south, are the two opposite 
points in which the celestial axis meets the starry concave. 
They are the poles of the celestial equator, or equinoctial, 
which is a great circle of the celestial sphere, in the same 
plane as the earth's equator. 

The celestial meridians are great circles of the celestial 
sphere passing through its poles, and intersecting the equi- 
noctial at right angles. When taken at intervals of 15° on 
the equinoctial, or any circle parallel to it, they form hour 
circles, since each of the heavenly bodies, in its apparent 
diurnal revolution, advances (360 H- 24— ) 15° in one hour. 

Shirts apparent Motion in the Ecliptic. 

(191.) The earth's annual revolution, from west to east, 
around the sun, causes an apparent revolution of the sun 
around the celestial sphere, in the same time, from west to 
east. These revolutions are in the plane of the Ecliptic, 
which is a great circle of the celestial sphere intersecting the 
equinoctial at an angle of about 23° 28'. 

The equinoxes, or equinoctial points, are the two points, 
180° asunder, in which the ecliptic intersects the celestial 
equator. The sun, in his apparent annual revolution in the 
ecliptic, arrives at one of these points about the 21st of 
March, whence that point is called the vernal equinox, and 
at the other about the 22d of September, whence that is called 
the autumnal equinox. At these times day and night are 
of equal length, in all parts of the world. 



SPHEKICAL TRIGONOMETRY. 197 

The Ecliptic is divided into 12 equal parts, called Signs ; 
each sign therefore contains 30°. The names of the signs, 
from west to east, beginning at the vernal equinox, are : 

1. Aries, 7. Libra, 

2. Taurus, 8. Scorpio, 

3. Gemini, 9. Sagittarius, 

4. Cancer, 10. Capricornus, 

5. Leo, 11. Aquarius, 

6. Virgo, 12. Pisces. 

Declination and Right Ascension y Celestial Latitude and 

Longitude. 

(192.) The Declination of any heavenly body is an arc of 
a meridian intercepted between the body and the equator. 
Its Right Ascension is an arc of the equator intercepted 
between the vernal equinox and a meridian passing through 
the body. 

The Latitude of any heavenly body is an arc of a secondary 
to the ecliptic intercepted between the body and the ecliptic. 
Its Longitude is an arc of the ecliptic intercepted between 
the vernal equinox and a secondary to the ecliptic passing 
through the body. 

Eight ascension and Longitude are reckoned in the order 
of the signs, through the entire circumference, or 360°. 

The Horizon and Circles related to it. 

(193.) The Horizon is the great circle of the celestial 
sphere which, to an observer on the general level of the 
earth's surface, is the boundary of the visible celestial hemi- 
sphere. 

The Zenith is the point of the celestial sphere which is 
vertically overhead to the observer ; and the Nadir is that 
point of the sphere which is diametrically opposite to the 
zenith. These two points are the poles of the horizon. 

Yertical Circles are great circles passing through the 
zenith and nadir of the place of observation ; being at right 
angles with the horizon. 



198 BOOK V. 

The Prime Vertical is the vertical circle which passes 
through the east and west points of the horizon ; being at 
right angles with the meridian of the place, which is a 
vertical passing through the north and south points of the 
horizon. 

Altitude — Azimuth — Amplitude. 

(194.) The Altitude of any heavenly body is the arc of a 
vertical circle intercepted between the body and the horizon. 

The Azimuth of any heavenly body is the arc of the 
horizon intercepted between the north or south point of the 
horizon and a vertical circle passing through the body. Its 
Amplitude is the arc of the horizon intercepted between the 
east or west point of the horizon and a vertical circle passing 
through the body ; being the complement of the azimuth. 

Relative Position of the Celestial Poles. 

(195.) To an observer on the general level of the earth's 
surface, one of the celestial poles is elevated above, and the 
other is depressed below, the horizon, just as many degrees as 
are contained in his latitude. 

Let C be the common centre 
of the earth and the celestial 
sphere, PP' the celestial axis, and 
EQ a diameter of the celestial 
equator. 

Let s be the observer's position 
on the earth, ZN a straight line 
joining his zenith and nadir, and 
$0 a straight line in the plane of 
his horizon. 

The observed elevation of the pole P will be the angle OsP. 
But the radius Cs of the earth is immeasurably small when 
viewed from the distance of the starry sphere ; so that the 
the lines Ys and Os would sensibly coincide with PC and 
OC ; and the elevation of the pole P is therefore sensibly 
equal to the angle OOP. 




SPHERICAL TRIGONOMETRY. 199 

The observer's latitude is the arc rs, which measures the 
angle rCs / and this angle is equal to OCP, since either of 
these two angles is the complement of the angle ZCP. The 
elevation of the pole P is therefore equal in degrees to the 
observer's latitude ; and the depression HOP of the opposite 
pole P' evidently contains the same number of degrees. 

In the diagrams accompanying the following Problems, we 
shall suppose the observer to be in the northern hemisphere ; 
in which situation the north pole of the heavens will be above 
his horizon. 



PROBLEM I. 

(196.) The Sun's Declination and Right Ascension given, 
to find his Longitude and the Obliquity of the Ecliptic. 

Let P and P' be the north and 
south poles of the celestial sphere, 
EQ the celestial equator, or equi- 
noctial, and E'C the ecliptic inter- 
secting the equator in the vernal 
equinox at the point A. 

The sun being always in the 
ecliptic, suppose him to be at S. 
Through the point S draw the me- 
ridian PSP', intersecting the equator 

in the point K. Then ES is the sun's declination, and AE 
his right ascension ; also AS is the sun's longitude (192). 

In the spherical triangle AES, right-angled at E, we have 
the sides ES and AE given, to find the side AS, which is the 
sun's longitude, and the angle EAS, which is the obliquity 
of the ecliptic to the equator. 

23. On the 17th of May, the sun's declination was observed 
to be 19° 15" 57", and his right ascension 53° 38' ; what was 
his longitude, and the obliquity of the ecliptic at the same 
time « Ans. 55° 57' 43" ; and 23° 27' 50J". 





200 BOOK 



PROBLEM n. 

(l 97.) The Sun's Declination given, to find the time of 
his Rising and Betting at any Place whose Latitude is 
known. 

Let HO be the horizon of the 
place, PEP'Q the meridian of the 
place, which is the celestial me- 
ridian passing through the north 
and south points of the horizon, 
S the sun in the horizon, which is 
his position at sunrise, and TST' 
the sun's apparent circuit on the 
given day. 

Draw the meridian PSP', through the sun, and cutting the 
celestial equator EQ in the point K; also let PAP' be a 
meridian at right angles with the meridian EPQ. 

The sun will be at S' on the meridian PAP' at six d clock; 
for this meridian evidently divides the 12 hours from mid- 
night, when the sun is at T, to midday, when he is at T', into 
two parts of 6 hours each. 

The time required for the sun to pass from S to S', sub- 
tracted from 6 hours, will give the time from midnight to 
sunrise, which is his time of describing the arc TS ; and the 
time of his describing SS / will be measured by the arc AR — 
allowing 15° to an hour. 360°-f- 24=15°. 

In the spherical triangle ARS, right-angled at R, the side 
RS, the sun's declination, will be given ; the angle RAS is 
measured by the arc QO, equal to PQ, 90°, — PO, and PO is 
equal to the given latitude of the place (195) ; the side AR 
may therefore be computed, and the time of sunrise be thence 
determined, as above explained. 

At sunset the sun is in the horizon, in the west ; the time 
would be that of his describing an arc equal to T'S'S, and 
would be measured by the equatorial arc ER, equal to 
90°+AR. 



SPHERICAL TRIGONOMETRY. 



201 



The time of the sun's setting would also be found by sub- 
tracting the time of his rising from 12 hours ; that is, by 
taking the time in which he would describe an arc equal to 
ST from the time in which he would describe T'T. Con- 
versely, the time of sunrise would be found by subtracting 
the time of sunset from 12 hours. 

24. At what time does the sun rise in latitude 52° 13' N"., 
when his decimation is 23° 28' 1ST. ? 

Ans. 3 o'clock, 43 ra., 46J s., A. M. 

PEOBLEM m. 



(198.)' The Latitude of the place and the Declination of a 
heavenly tody given, to find the Altitude and Azimuth of 
the hody when on the six o'clock hour circle. 

Let HO be the horizon of the 
place, Z the zenith, PEP'Q the 
meridian of the place, and S the 
sun on the six o'clock hour circle 
PAP', which is a meridian at 
right angles with the meridian 
EPQ. 

Draw the vertical circle ZSIS", 
cutting the horizon in the point 
B, and the prime vertical ZAE", 
cutting the horizon in the point A. 

In the spherical triangle ABS, right-angled at B, the angle 
BAS is measured by the arc OP, which is equal to the given 
latitude (195), and the side AS is the sun's given declination. 
The sides BS and AB may therefore be computed. BS is 
the sun's required altitude ; AB is the sun's amplitude, and 
ABO, 90°,— AB, gives BO, his required azimuth (194). 




25. What were the altitude and azimuth of the star Arc- 
turus, when upon the six o'clock hour circle of Greenwich, 
latitude 51° 28' 40" K, on the 1st of April ; its declination 

Ans. 15° 36' 27" ; and 77° 9' 5". 



being 20° 6' 50" K ? 



202 BOOK V. 



PROBLEM IV. 




(199.) The Latitudes and Longitudes, or the Declinations 
and Right Ascensions, of two heavenly bodies given, to find 
their Distance from each other. 

Let A and B be two stars whose 
latitudes and longitudes are given. 

Let EC be the ecliptic, and P and 
P' its poles. Draw the secondaries 
to the ecliptic PAP' and PBF, 
meeting that circle in the points d 
and o / then Ad and Bo are the re- 
spective latitudes of the two stars. 

Draw AB, an arc of a great circle. In the spherical tri- 
angle ABP, the side PA is equal to Yd, 90°,+ the given 
latitude Ad ; and PB is equal to To, 90°, — the given latitude 
B<9. The angle APB is measured by the arc do, which is 
the difference of longitude of the two stars. The side AB, 
which measures the distance between the stars may there- 
fore be computed (182). 

If the declinations and right ascensions of the two stars 
were given, instead of their latitudes and longitudes, the 
procedure would be entirely similar for finding the two sides 
and their included angle APB— the equator taking the place 
of the ecliptic in the diagram. 

26. Find the angular distance between the stars Procyon 
and Capella, the latitude of the former being 15° 58' 14" S., 
its longitude 112° 55' 42" ; the latitude of the latter 22° 51' 
57" K, and its longitude 78° 57' 57". Ans. 51° 6' 39". 

27. Find the angular distance between the stars Procyon 
and Sirius, the declination of Procyon being 5° 45' 3" K, 
its right ascension 112° 6' 47", the declination of Sirius 16° 
26' 35" S., and its right ascension 99° 0' 21". 

Ans. 25° 42' 13". 



SPHERICAL TRIGONOMETRY. 



203 



PROBLEM V. 




(200.) The Latitude of the place, and the sun's Declination 
and Altitude, given, to find the Hour of the Day. 

Let HO be the horizon, and 
PEP'Q the meridian, of the place, 
P and P' the poles of the equator 
EQ, and Z the zenith. 

Let S be the place of the sun. 
Draw the vertical circle ZSN", cut- 
ting the horizon in B, and the 
meridian or hour circle PSP', cut- 
ting the equator in A. 

In the spherical triangle SPZ, the side SP is equal to AP, 
90°— AS, the sun's given declination; the side PZ is equal 
to OZ, 90°,-OP, equal to the given latitude (195) ; and the 
side SZ is equal to BZ, 90°— BS, the sun's given altitude. 
The angle SPZ may therefore be computed (184). 

This angle reduced to time, by allowing 15° to an hour, 
gives the number of hours before, or after, mid-day when the 
sun is at S, according as the sun is east, or west, of the 
meridian. 

28. What was the hour of the day, in latitude 39° 54' K, 
when the sun's declination was 17° 24' 7" 1ST., and altitude 
15° 53' 40" ; the observation having been made when the 
sun was west of the meridian ? 

Ans. 5 o'clock, 34 m., 16 s., P. M. 



PROBLEM VI. 



(201.) The Latitude of the place and the sun's Declination 
given, to find the Beginning and end of Twilight. 

Twilight begins in the morning, and ends in the evening, 
when the sun is 18° below the horizon, measured on a vertical 
circle passing through the sun. 



204 



BOOK V. 




Let HO be the horizon, and 
PEP'Q the meridian of the place, 
P and P' the poles of the equator 
EQ, and S the sun's place at the 
beginning, or end, of twilight. 

Draw the vertical circle ZSN", 
cutting the horizon in <?, and the 
meridian PSP', cutting the equa- 
tor in r. 



In the spherical triangle SPZ, the side SP is equal to rP, 
90°,— rS, the sun's given declination ; the side PZ is equal 
to OZ, 90°,— OP, equal to the given latitude (195) ; and the 
side SZ is equal to cZ, 90°,+cS, 18°, equal to 108°. The 
angle SPZ may therefore be computed (184). 

This angle reduced to time, by allowing 15° to an hour, 
gives the number of hours before mid-day when twilight 
begins in the morning, and the number after mid-day when 
twilight terminates in the evening. 



29. At what time does the morning twilight begin in lati- 
tude 55° 57' 20" K, when the sun's declination is 12° 38' 9" 
K ? Ans. 1 o'clock, 44 m., 40f §., A. M. 

30. At what time does the evening twilight end in latitude 
52° 12' 35" K, when the sun's declination is 15° 55' 25" K ? 

Ans. 10 o'clock, 12 m., 40 s., P. M. 



MISCELLANEOUS PROBLEMS. 



205 



MISCELLANEOUS PROBLEMS 

IN THE APPLICATIONS OF TEIGOIOIETET. 

The computations requisite for obtaining the Answers to 
the following Problems are such as the student is here sup- 
posed to be familiar with ; it is intended that he be required 
only to show the methods of solution, without the numerical 
operations. 



EXAMPLE. 

From a point within an equilateral triangle, I measured 
the distances to the vertices of the three angles, and found 
them to be 10, 12, and 15 poles, respectively. Required the 
sides of the triangle. 

Construction. — With the 
three given distances construct 
the triangle ABC (Geom.278); 
on any one of its sides, as AC, 
construct the equilateral tri- 
angle ACD ; draw the straight 
line BD, and upon it construct 
the equilateral triangle BDF ; 
this will be the required tri- 
angle, and C the point from 
which the three distances were 
measured to its vertices. 

For, by the construction, CB is one of those distances, and 
CD is equal to another, AC, the triangle ACD being equila- 
teral ; also the angles ADC and BDF are equal, each of them 
being i of two right angles, and hence the angles ADB and 
CDF are equal ; so that the triangles ADB and CDF have 
the two sides AD and DB and the included angle of the one 
equal to the two sides CD and DF and the included angle of 
the other ; and therefore the distance CF is equal to the 
remaining given distance AB (Geom. 45). 




206 BOOK V. 

Solution. — In the triangle ABC the three sides are given ; 
find the angle ACB (58). In the equilateral triangle ACD 
the angle ACD is | of 180°, that is, 60°. The angle BCD is 
the sum of the two angles ACB and ACD ; then in the 
triangle BDC we shall have the two sides BC and CD, and 
their included angle, to find BD, one of the equal sides of 
the triangle BDF (57). 

1. At the distance of 170 feet from the base of a tower, 
standing on a horizontal plane, the angle of elevation of its 
summit was measured, and found to be 52° 30'. What was 
the height of the tower? 

2. A person on one side of a river, observed that the angle 
of elevation of the top of an obelisk on the other side was 
35° 54'. Going 100 feet farther from the obelisk, he found 
the elevation of its top to be 20° ; what was the height of 
its top above the horizontal plane of observation ? 

3. Erom the summit of a rock which rises 150 feet above 
the margin of a stream of water, the angle of depression of 
the opposite margin was measured, and found to be 41° 48'. 
Required the breadth of the stream. 

4. Wishing to determine the height of a steeple standing 
on an inclined plane, I measured 200 from its base, and there 
found the elevation of its summit to be 47° 50'. Measuring 
80 feet farther, I then found the elevation of its summit to 
be 38° 30' ; what was the height of the steeple ? 

5. Wishing to know the extent of a piece of water, or the 
distance between two headlands, I measured from each of 
them to a certain point inland, and found the two distances 
to be 735 yards and 840 yards; also the horizontal angle 
contained between these two lines was 55° 40'. Requirec" 
the distance between the two headlands. 

6. At the top of a castle 60 feet in height, near the sea 
shore, the angle of depression of a ship at anchor was 4° 52', 



MISCELLANEOUS PROBLEMS. 207 

and at the base the ship's depression was 4°. Bequired the 
horizontal distance to the ship, and the height of the castle's 
base above the level of the sea. 

7. The distances of three objects from one another are, 
AB 12 miles, AC 8 miles, and BC 7|- miles. At the station 
D, in the straight line CA produced, the horizontal angle 
ADB was 17° 47' 19" ; what were the distances from that 
station to each of the objects ? 

8. From the top of a mountain three miles in height, the 
angle of depression of the remotest visible point of the earth's 
surface was taken, and found to be 2° 13 / 27". It is required, 
from these data, to compute the diameter of the earth, on the 
supposition of its being a perfect sphere. 

9. Wishing to know the distance between a church at A 
and a tower at B, I took the angles ADB 72° 30', ADO 
89°, and BDE 54° 30'. I then measured DC 200 yards, 
and took the angle DCA 50° 30'; finally, I measured DE 
200 yards, and took the angle DEB 88° 30'. Eequired the 
distance AB. 

10. A point of land was observed, from a ship at sea, 
to bear east by south • and after sailing north-east 12 
miles, it was found to bear S. E. by E. It is required 
to determine the ship's distance from it at the last observ- 
ation. 

11. The side AB of a triangular piece of ground runs 1ST. 
35° E., Z chains, and the side AC runs S. 80° E., 40 chains. 
Find the bearing and the length of the side BC. 

12. With the same data as in Problem 11, find the bearing 
and the length of a straight line AD, running from the point 
A, so that the point D shall be equidistant from the points 
A, B, and C (Geom. 238). 



10 



208 BOOK V . 

13. One side of a parallelogram runs S. S0}° E., 18.5 chains 
and, at the same station, the adjacent side runs K. 60^° E. 
40 chains. What is the area of the parallelogram ? 

14. One side of a triangular field runs N. 21 J° E., 36 chains 
and, at the same station, another side runs 1ST. 8° 30' W. 
42.75 chains. What is the area of the field ? 

15. A triangular tract of land, ABC, has the angle A 54° 
25', the angle B 60° 10', and the side AC 34.75 chains. 
What is the area of the tract ? 

16. The two diagonals of a quadrangular piece of ground 
measure 75 chains, and 83.5 chains, respectively, and they 
intersect each other at an angle of 45° 44'. What is the area 
of the quadrangle ? 

17. In a quadrilateral field ABCD, the side AB is 20 poles, 
BC 16.5 poles, CD 30 poles ; also the angles A, B, and C are 
85°, 94° 25', and 120°, respectively. What is the area of the 
quadrilateral ? 

18. A flower garden is laid out in the form of a regulai 
pentagon whose vertices are in the circumference of a circle 
of 40 rods in diameter. What is the area of the garden ? 

19. The side AB of a tract of land ran through a thicket, 
so that its bearing and length could not be directly measure 

I therefore ran AC, K 10*° E., 4 chains, CD, K 20° 1 
16 chains, and DB, S. 40° E., 20 chains. From these da 
find the bearing and length of AB. 

20. The road from A to B runs S. 12° W., 75 chains ; 
thence S. 3}° W., 36 chains ; thence S. 30J° E., 100 chains. 
What would be the bearing, and the length, of a direct road 
from A to B ? 



MISCELLANEOUS PROBLEMS. 209 

21. The interior angles and the sides of a tract of land, 
ABCDF, are as follows : 

the angle A 72° 42', the side AB 30 chains ; 
the angle B 130° , the side BC 10 chains ; 
the angle C 131° , the side CD 30 chains ; 
the angle D 100° , the side DF 20 chains ; 
the angle F 106° 18', the side FA 39.42 ch. 
"What is the area of this tract ? 



22. In the triangle ABO the angle A is 48°, the side AB 
12 chains, and the side AC 16 chains. It is required to cut 
off 3 acres, towards the angle A, by a straight line BD drawn 
from the point B to the side AC ; what must be the length 
of AD? 

23. With the same data as in Problem 22, it is required to 
cut off 4 acres from the given triangle, towards the angle B, 
by a straight line DF drawn from the side AB to BC, and 
parallel to AC. "What must be the length of BD ? 

24. The side AB of a tract of land runs K 40° E., and the 
adjoining side AC runs S. 80° E. It is required to cut off 
the triangular piece ADF which shall contain 5 acres, by a 
straight line DF running from AC to AB, the distance AD 
to be 16 poles ; what must be the bearing of the line DF ? 

25. With the same data as in Problem 24, it is required 
to cut off a triangular piece AG-H containing 6 acres, by a 
straight line GH running !N". 2° W. from a point G in the 
line AC. What must be the length of AG ? 

26. The side AB of a tract of land runs K 43° E., and the 
adjoining side AC runs due east. It is required to cut off a 
triangular field ADF containing 4 acres, by a straight line 
DF running from AC to AB, in such direction as to be the 



210 BOOK V. 

shortest line that will cut off said area. What must be the 
length of AD ? and the bearing of DF ? 

27. With the same data as in Problem 26, it is required to 
cut off a triangular field AGH containing 6 acres, by a 
straight line GH running from a point G in the line AC, 
through the point P, which bears K. 65° E., 10 chains, from 
the point A; what must be the length of AG ? and the bear- 
ing of GH ? 

28. The side AB of a tract of land runs S. 75° E., the side 
AC runs N. 5° E., 40 chains, and the side CD is parallel to 
AB. It is required to cut off 10 acres, by a line FG running 
from a point F in the side AB, and parallel to AC ; what 
must be the distance AF ; 

29. Having the same data as in Problem 28, it is required 
to cut off 12 acres, by a line KL running N". 3° W. from a 
point K in the side AB. What must be the distance AK? 

30. The side AB of a tract of land runs due East, the side 
AC runs !N". 5° E., 16 chains, the side CD is parallel to AB. 
It is required to cut off 16 acres, by a line FG running from 
a point F in the side AB, through the point P which bears 
IS". 30° E., 20 chains from the point A ; what must be the 
distance AF ? 

31. The side AB of a tract of land runs S. 85° E., the side 
AC runs 1ST. 10° E., 13 chains, the side CD runs 1ST. 64° E. 
It is required to cut off 10 acres, by a line KL running due 
North, from a point K in the side AB. What must be the 
distance AK? 

32. With the same data as in Problem 31, it is required to 
cut off 25 acres, by a straight line FG running from AB, in 
such direction as to be the shortest line that will cut off said 
area. What must be the length of AF % and the bearing of 
FG? 



MISCELLANEOUS PROBLEMS. 211 

33. The side AB of a tract of land runs due East, the side 
AC runs K 10° E., 35 chains, the side CD runs 1ST. 45° E. 
It is required to cut off 20 acres, by a straight line EX run- 
ning from a point K in the side AB, through the point P, 
which bears N. 50° E., 18 chains from the point A. What 
must be the distance AK ? and the bearing of KX ? 

34. The side AB of a tract of land runs S. 80° E., the side 
AC runs 1ST. 10°, 12 chains, the side CD runs 1ST. 45° E., 
20 chains, the side DE runs N". 75° E. It is required to cut 
off 30 acres, by a straight line PI running from AB, the dist- 
ance AP to be 15 chains ; what must be the bearing of the 
line PI? 

35. Having the same data as in Problem 34, it is required 
to cut off 33 acres by a straight line G-H running N. 10° E. 
from a point Gr in the side AB ; what must be the distance 
AG? 



36. A ship in latitude 37° 10' 1ST. is bound to a port in 
latitude 33° E". which lies 180 miles west of the meridian of 
the ship ; but by reason of contrary winds, she sails the 
following courses, viz. : S. W. by W., 27 miles, W. S. "W., 
30 miles, "W. by S., 25 miles, W. by 1ST., 18 miles. Kequired 
the latitude the ship has reached her distance from the meri- 
dian left, and her course and distance to her intended port. 

37. If a ship sail due East 126 miles, from the North Cape, 
in latitude 71° 10' ]ST., and then due North till she reaches 
the latitude 73° 26' N. ; how far must she sail West to reach 
the meridian of the North Cape ? 

38. Two ships on the parallel of 40° K, have 10° difference 
of longitude, and they both sail directly South, a distance of 
500 miles. Required their distance from each other on the 
parallel left, and on the parallel arrived at. 



212 BOOK V. 

39. A ship sailed from latitude 34° 30' K, and longitude 
10° W., on a S. W. course, 500 miles. What latitude and 
longitude did the ship reach ? 

40. On what course, continued invariably, must a ship 
sail from Cape St. Eoque, in latitude 5° 28' S., longitude 
35° IT W., to arrive at the Cape of Good Hope, in latitude 
34° 22' S., longitude 18° 24' E. ? and what would be the 
distance sailed ? 

41. With the same data as in Problem 40, what is the 
distance between Cape St. Boque and the Cape of Good 
Hope, on a great circle of the earth ? and on what course 
must a ship sail from the former place in order to reach the 
latter by the method of great-circle sailing % 

42. What would be the course, and the distance, on a 
rhumb line extending along the level surface of the earth, 
from Savannah, in latitude 32° 4' 56" 1ST., longitude 91° V 9" 
W., to Cincinnati, in latitude 39° 5' 54" K, longitude 84° 
24' W.? 

43. With the same data as in Problem 42, what is the 
shortest distance, on the level surface of the earth, between 
Cincinnati and Savannah ? and on what course must a road 
leave the former city in order to reach the latter with that 
distance ? 

44. The declination of the star Aldebaran is 16° 8' 36" K, 
and its right ascension is 60° 25' 43". What is the angular 
distance between that star and the Sun, at the moment of 
the vernal equinox f 

45. How long is the Sun above the horizon, in the latitude 
of New York city, 40° 42' N"., on the longest day in the year, 
allowing the sun's declination at the time to be 23° 27' 36.5"? 
and what is the duration of twilight in the same latitude, at 
that time? 



TABLE I. 



CONTAINING 



LOGARITHMS OE NUMBERS 



Fbom 1 to 10,000. 



N. 


Log. 


N. 


Log. 


N. 


Log. 


N. 


Log. 


1 


0* 000000 


26 


I-4I4973 


51 


1-707570 


76 


1 -880814 


2 


o-3oio3o 


27 


i-43i364 


52 


1 -716003 


77 


1-886491 


8 


0-477121 


28 


1 -447i58 


53 


1-724276 


78 


1-892095 


4 


o« 602060 


29 


1-462398 


54 


1-732394 


79 


1-897627 


5 


0-698970 


30 


I-477 121 


55 


i-74o363 


80 


1-903090 


6 


0-778151 
0-845098 


31 


i-49i362 


56 


1-748188 


81 


1-908485 


7 


32 


I.5o5i5o 


57 


1-755875 


82 


1-913814 


8 


0-903090 


33 


i-5i85i4 


58 


1-763428 


83 


1-919078 


9 


0-954243 


34 


1 -53 1479 


59 


1-770852 


84 


1-924279 


10 - 


I -000000 


35 


1 • 544068 


60 


i-778i5i 


85 


1-929419 


11 


i-o4i3o3 

1-079181 


36 


i-5563o3 


61 


i-78533o 


86 


1 -934498 


1-2 


37 


1-568202 


62 


1-792392 


87 


1-939519 
1-944483 


13 


i-ii3943 


38 


1-579784 


63 


1-799341 


88 


14 


1-146128 


39 


i- 591065 


64 


1-806180 


89 


1-949390 


15 


1-176091 


40 


1 -602060 


65 


1-812913 


90 


1-954243 


16 


1-204120 


41 


1-612784 


66 


1-819544 


91 


1-959041 


17 


i- 230449 
1-255273 


42 


1-623249 
1-633468 


67 


1-826075 


92 


1-963788 


18 


43 


68 


i-8325og 


93 


1-968483 


19 


1-278754 


44 


1-643453 


69 


1-838849 
1-845098 


94 


1-973128 


20 


i-3oio3o 


45 


1 -6532i3 


70 


95 


1-977724 


21 


1*322219 


46 


1-662758 


71 


i-85i258 


96 


1-982271 


22 


1-342423 


47 


1-672098 


72 


i-85 7 333 


97 


1-986772 


23 


1-361728 


48 


1-681241 


73 


1-863323 


98 


1-991226 


24 


i-38o2ii 


49 


1-690196 


74 


1-869232 


99 


1-995635 


I 25 


1-397940 


50 


1 • 698970 


75 


1-875061 


100 


2-000000 



Remark. — In the following Table, the first two figures, in the first column of 
Logarithms, are to be prefixed to each of the numbers, in the same horizontal 
line, in the next nine columns; but when a point (•) occurs, a is to be put 
in its place, and the two initial figures are to be taken from the next line below. 



LOGARITHMS OF NUMBERS. 



11. 





1 


! 2 


3 


4 


5 


6 


7 


8 


9 


B, 1 


100 


oooooo 


0434 


0868 


i3oi 


17^4 


2166 


2D98 


3029 


346i 


38 9 i 


432 i 


101 


4321 


475i 


5i8i 


5609 


6o38 


6466 


6894 


7321 


7748 


8174 I 428 | 


1 102 


86oo 


9026 


9451 


9876 


•3oo 


•724 


1147 


1D70 


i 99 3 


24i5 


424 


, 103 


012837 


3269 


368o 


4100 


4521 


4940 


536o 


5779 


6197 


6616 


419 


104 


7033 


745i 


7868 


8284 


8700 


91 16 


9532 


9947 


•36 1 


• 77 5 


416 ; 


105 


021189 


i6o3 


2016 


2428 


2841 


3252 


3664 


4075 


4486 


4896 


412 • 


106 


53o6 


5715 


6i25 


6533 


6942 


7350 


77^7 


8164 


85 7 i 


8978 


408 ; 


107 


9384 
o33424 


9789 
3826 


•195 


•600 


1004 


1408 


1812 


2216 


2619 


302I 


404 


108 


4227 


4628 


5029 


5430 


583o 


623o 


6629 


7028 


400 


109 


7426 


7825 


8223 


8620 


9017 


9414 


9811 


•207 


•602 


•998 


396 ! 


110 


041393 


1787 


2182 


2576 


2969 


3362 


3 7 55 


4i48 


4540 


4932 


3 9 3 j 


111 


5323 


5714 


6io5 


6495 


6885 


7275 


7664 


8o53 


8442 


883o 


38 9 f 


112 


9218 


9606 
3463 


9993 


•38o 


•766 


u53 


i538 


1924 


23og 


2694 


386 ; 


113 


053078 


3846 


423o 


46 1 3 


4996 


5378 


5700 


6142 


6624 


382 ■ 1 
379 I 


114 


6905 


7286 


7666 


8046 


8426 


88o5 


9i85 


9563 


9942 


•320 


115 


060698 


1075 


1452 


1829 


2206 


2582 


2 9 58 


3333 


3709 


4o83 


376 j 


116 


4458 


4832 


52o6 


558o 


5 9 53 


6326 


6699 


7071 


7443 


7815 


372 ' 


117 


8186 


8557 


8928 


9298 


9668 


••38 


•407 


• 77 6 


1145 


»5H 


369 1 


118 


071882 


225 O 


2617 


2985 


3352 


3718 


4o85 


445 1 


4816 


5i82 


366 i 


119s 


5547 


5912 


6276 


6640 


7004 


7368 


773i 


8094 


8457 


8819 


363 | 


120 


079181 


9543 


99°4 


•266 


•626 


•987 


i347 


1707 


2067 


2426 


36o ' 


121 


082785 


3i44 


35o3 


386 1 


4219 


4076 


4934 


0291 


5647 


6004 


357 I 


122 


636a 


6716 


7071 


7426 


7781 


8i36 


8490 


8845 


9198 


9552 
S071 


355 


123 


9905 


•258 


•611 


• 9 63 


i3i5 


1667 


2018 


2370 


2721 


35 1 > 


124 


093422 


3772 
7 25 7 


4122 


4471 


4820 


5i6g 


55i8 


5866 


62i5 


6562 


349 * 


125 


6910 


7604 


79 5 1 


8298 


8644 


8990 
2434 


9335 


9681 


••26 


346 


126 


100371 


0715 


1009 


i4o3 


1747 


2091 


2777 


3i 19 


3462 


343 > 


127 


38o4 


4146 


4487 


4828 


5169 
8565 


55io 


585i 


61 91 


653i 


6871 


340 ; 


128 


7210 


7549 


7888 


8227 


8 9 o3 


9241 


9579 


9916 


•253 


338 ; 


129 


110690 


0926 


1263 


1 5 99 


1934 


2270 


26o5 


2940 


3275 


3609 


335 


130 


113943 


4277 


4611 


4944 


5278 


56n 


5943 


6276 


6608 


6940 


333 f 


131 


7271 


7603 


7934 


8265 


85 9 5 


8926 


9256 


9586 


9915 


•245 


33o 


132 


120574 


0903 


123l 


i56o 


1888 


2216 


2544 


287I 


3i 9 8 


3525 


3a8 ! 


133 


3852 


4178 


45o4 


483o 


5i56 


5481 


58o6 


6i3i 


6456 


6781 


325 


134 


7io5 


7429 
o655 


7753 


8076 


8399 


8722 


9045 


9 368 


9690 


••12 


3z3 | 


135 


i3o334 


0977 


1298 


1619 


1939 


2260 


258o 


2900 


3219 
64o3 


321 • 


136 


3539 


3858 


4177 


4496 


4814 


5i33 


545 1 


5769 


6086 


3i8» | 


137 


6721 


7037 


7354 


7671 


79 8 7 


83o3 


8618 


8 9 34 


9249 


9064 


3*5 


138 


9879 


•i94 


•5o8 


•822 


1 1 36 


i45o 


i 7 63 


2076 


238 9 


2702 


3*4 
3*1 \ 

i 


139 


i43oi5 


3327 


3639 


3951 


4263 


4574 


4885 


5196 


55©7 


58i8 


140 


146128 


6438 


6748 


7o58 


7 36 7 


7676 


7985 


8294 


86o3 


891 1 


309 [ 


141 


9219 


9527 


9835 


•142 


•449 


• 7 56 


io63 


1370 


1676 


1982 


307 


142 


152288 


2594 


2900 


3205 


35io 


38i5 


4120 


4424 


4728 


5o32 


3o5 


143 


5336 


564o 


5943 


6246 


6549 


6852 


7i54 


7457 


7759 


8061 


3o3 j 


144 


8362 


8664 


8 9 65 


9266 


9 56 7 


9868 


•168 


•469 


•769 


1068 


3oi 


145 


i6i368 


1667 


1967 


2266 


2564 


2863 


3i6i 


3460 


3758 


4o55 


299 


146 


4353 


465o 


4947 


5244 


554i 


5838 


61 34 


643o 


6726 


7022 


297 


147 


7 3i 7 


7613 


7908 


8203 


8497 


8792 


9086 


938o 


9674 


9968 


2 9 5 


148 


170262 


o555 


0848 


1141 


1434 


1726 


2019 


23 1 1 


26o3 


2895 


293 


149 
150 


3i86 


3478 


3769 


4060 


435 1 


4641 


4g32 


5222 


55i2 


58o2 


291 


176091 


638i 


6670 


6959 


7248 


7 536 


7825 


8n3 


8401 


8689 


289 1 


151 


8977 


9264 


9552 


9 83 9 


•126 


•4i3 


•699 


•985 


1272 


i558 


287 i 


152 


181844 


2129 

4975 


24i5 


2700 


2 9 85 


3270 


3555 


383 9 


4123 


4407 


285 ; 


153 


4691 


6259 


5542 


58a5 


6108 


6391 


6674 


6 9 56 


7 23 9 


2>83 : 


154 


7521 


7803 


8084 


8366 


8647 


8928 


9209 


9490 


9771 


©•5i 


281 ; 


155 


190332 


0612 


0892 


1171 


U5i 


1730 


2010 


2289 


2567 


2846 


279 1 


156 


3i25 


34o3 


368i 


3959 


4237 


45i4 


4792 


5069 


5346 


5623 


278 


157 


5899 


6176 


6453 


6729 


7oo5 


7281 


7556 


7832 


8107 


8382 


276 1 


158 


8657 


8 9 3 2 


9206 


9481 


97 55 


••29 


•3o3 


•5 77 
33o5 


•85o 


1 1 24 


274 | 
272 j 


159 


201397 


167a 


1943 


22 1 & 


2488 


2761 


3o33 


3577 


3848 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D.. 



L0GAE1THMS OF NTTMBEKS. 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


160 


204120 


4391 


4663 


4934 


5204 


5475 


5746 


6016 


6286 


6556 


271 


161 


6826 


7096 
97 83 


7365 


7634 


7904 


8i 7 3 


8441 


8fio 


8979 


9247 


269 


162 


9Di5 


••5i 


•319 


•586 


•853 


1121 


1 388 


1 654 


1921 
4579 


267 
266 | 


163 


212188 


2454 


2720 


2986 


3252 


35i8 


3 7 83 


4049 


43 1 A 


164 


4844 


5109 


53 7 3 


5638 


5902 


6166 


643o 


6694 


6957 


7221 


264 


165 


7484 


7747 


8010 


8273 


8536 


8798 


9060 


9323 


9585 


9846 


262 


166 


220108 


0370 


o63i 


0892 


n53 


i4U 


i6 7 5 


1936 


2196 


2456 


261 


167 


2716 


2976 


3236 


3496 


3 7 55 


4oi5 


4274 


4533 


4792 


5o5i 


259 
258 


168 


53oo 


5568 


5826 


6084 


6342 


6600 


.6858 


7!i5 


7372 


763o 


169 


7887 


8144 


8400 


8657 


8 9 i3 


9170 


9426 


9682 


9938 


•193 


256 


170 


23o449 


0704 


0960 


I2l5 


1470 


1724 


1979 


2234 


2488 


2742 


254 


171 


2996 
5528 


325o 


35o4 


3757 


401 1 


4264 


45i7 


4770 


5o23 


5276 


253 


172 


5 7 8i 


6o33 


6285 


6537 


6789 


7041 


7292 


7544 


77 9 5 


252 


173 


8046 


8297 


8548 


8799 


9049 


9299 


q55o 


9800 


•®5o 


•3oo 


25o 


174 


240549 


0799 


1048 


1297 


i546 


i 79 5 


2044 


2293 


254i 


2790 


249 


175 


3o38 


3286 


3534 


3782 


4o3o 


4277 


4525 


4772 


5019 


5266 


248 ' 


176 


55i3 


5 7 5 9 


6006 


6252 


6499 


6745 


6991 


7237 


7482 


7728 


246 [ 


177 j 


7973 


8219 


8464 


8709 


8954 


9198 


9443 


9687 

2125 


99 32 


•176 


245 1 


178 


250420 


0664 


0908 
3338 


1 i5i 


1395 


i638 


1881 


2368 


2610 


243 


179 j 


2853 


3096 


358o 

5996 
83 9 8 


3822 


4064 


43o6 


4548 


4790 


5o3i 


242 | 


180 


255273 


55i4 


5 7 55 


6237 


6477 


6718 


6 9 58 


7198 


7439 


241 


131 


7679 


7918 


8i58 


863 7 


8877 


9116 


9355 


9594 


9 833 


23 9 [ 


182 | 


260071 


o3io 


0548 


0787 


1025 


1263 


i5oi 


1739 


1976 


2214 


238 


183 i 


245i 


2688 


2925 


3i62 


3399 


3636 


38 7 3 


4109 


4346 


4582 


23 7 


184 


4818 


5o54 


5290 


5525 


5761 


5996 


6232 


6467 


6702 


6937 


235 


185 h 


7172 


.7406 


7641 


7875 


8110 


8344 


85 7 8 


8812 


9046 


9279 


234 


186 


9 5i3 


9746 


9980 


•2l3 


•446 


•679 


•912 


ii44 


1377 


1609 


233 


187 


271842 


2074 


23o6 


2538 


2770 


3ooi 


3233 


3464 


36 9 6 


3 9 2 7 


232 


188 


4i58 


438 9 


4620 


485o 


5o8i 


53u 


5542 


5 77 2 


6002 


6232 


23o 


189 


6462 


6692 


6921 


7 i5i 


7380 


7609 


7 838 


8067 


8296 


8525 


229 


190 


278754 


8982 


921 1 


9439 


9667 


9895 


•123 


•35i 


•5 7 8 


•806 


228 


19.1 


28io33 


1261 


1488 


171 5 


1942 


2169 


2396 


2622 


2849 


3075 


227 | 


192 1 


33oi 


3527 


3753 


3979 


42o5 


443 1 


4656 


4882 


5107 


5332 


226 I 


193 I 


5557 


5782 


6007 


6232 


6456 


6681 


6905 


7i3o 


7354 


7 5 7 '8 


225 I 


194 | 


7802 


8026 


8249 


8473 


8696 


8920 


9U3 


9366 


9589 


9812 


223 | 


195 ! 


290035 


0257 


0480 


0702 


0925 


1 147 


1369 


1591 


i8i3 


2o34 


222 1 


196 j 


2256 


2478 


2699 


2920 


3i4i 


3363 


3584 


38o4 


4025 


4246 


221 I 


197 i 


4466 


4687 


4907 


5i 27 


5347 


5567 


5787 


6007 


6226 


6446 


220 1 


198 | 


6665 


6884 


7104 


7323 


7542 


7761 


7979 


8198 


8416 


8635 


219 I 


199 
200 


8853 


9071 


9289 


9 5o 7 


9725 


9940 


•161 


•3 7 8 


•595 


•8i3 


218 [ 


3oio3o 


1247 


1464 


1681 


1898 


2114 


233i 


2547 


2764 


2980 


217 


20! 


3196 


3412 


3628 


3844 


4059 


4275 


4491 


4706 


4921 


5i36 


1 


20-2 


535i 


5566 


5 7 8i 


5996 


6211 


6425 


663 9 


6854 


7068 


72S2 


2 I 5 1 


203 


7496 


77io 


7924 


8i3 7 


835i 


8564 


8778 


899.. 


9204 


9417 


2,3 


204 J 


9630 


9843 


••56 


•268 


•481 


•6 9 3 


•906 


1118 


i33o 


i542 


212 1 


205 


311754 


1966 


2177 


238 9 


2600 


2812 


3o23 


3234 


3445 


3656 


211 1 


206 


3867 


4078 


4289 


4499 


4710 


4920 


5i3o 


5340 


555 1 


5760 


210 I 


207 | 


5970 


6180 


6390 


6599 


6809 


7018 


7227 


7436 


7646 


7854 


200 1 


20S 


8o63 


8272 


8481 


8689 


8898 


9106 


93i4 


9 522 


973o 


9938 


208 I 


209 


320146 


o354 


o562 


0769 


0977 


1 184 


1391 


i5 9 8 


i8o5 


2012 


207 J 


210 


322219 


2426 


2633 


283g 


3o46 


3252 


3458 


3665 


3871 


4077 


206 | 


211 


4282 


4488 


4694 


4899 


5io5 


53 10 


55i6 


5721 


5926 


6i3i 


205 J 


212 


6336 


654i 


6745 


6950 


7i55 


7359 


7563 


7767 


7972 


8176 


204 1 


213 


838o 


8583 


8787 


8991 


9194 


9398 


9601 


9805 


•••8 


•21 1 


203 p 


214 


! 33o4U 


0617 


0819 


1022 


1225 


1427 


i63o 


i832 


2o34 


2236 


202 jj 


215 


2438 2640 


2842 


3o44 


3246 


3447 


3649 


385o 


4o5i 


4253 


202 1 


216 


4454 1 4655 


4856 


5o57 


5237 


5458 


5658 


585 9 
7858 


6059 


6260 


201 j 
200 


217 


6460 ! 6660 


6860 


7060 


7260 


745 9 


7659 


8o58 


8207 


21S 


8456 i 8656 | 8855 


9054 


9253 


945 r 


9600 


9849 


••47 


•246 


I99 


219 


1 340444 0642 | 0841 


1039 


1237 


1435 


1632 


i83o 


2028 


2225 


19 * 


N. 


~°~ 


1 


2 


3 


4 


5 


6 


7 


8 


9 


B. 



10* 



LOGARITHMS OF NUMBERS. 



F. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


1 


220 


342423 


2620 


2817 


3oi4 


3212 


3409 


36o6 


38o2 


3999 


4196 

6i5 7 


197 


221 


4392 


458g 


4785 


4981 


5l 7 8 


53 7 4 


5570 


5 7 66 


5962 


196 


222 


6353 


6549 


6744 


6939 


7135 


733o 


7525 


7720 


79 i5 


8110 


i 9 5 


223 


83o5 


85oo 


8694 


8889 


9 o83 


9278 


9472 


9666 


9860 


••54 


194 


224 


35o248 


0442 


o636 


0829 


1023 


1216 


1410 


i6o3 


1796 


1989 


i 9 3 


225 


2i83 


23 7 5 


2568 


2761 


2954 
4876 


3i47 


333 9 


3532 


3724 


3 9 i6 


I9 3 


226 


4108 


43oi 


4493 


4685 


5o68 


526o 


5452 


5643 


5834 


192 


227 


6026 


6217 


6408 


6599 


6790 


6981 
8886 


7172 


7363 


7554 


7744 


191 


228 


7o35 
9 835 


8i25 


83i6 


85o6 


8696 


9076 


9266 


9456 


9646 


IS 


229 
230 


••25 


•2l5 


•404 


•593 


• 7 83 


•972 


1 161 


i35o 


1539 


361728 


1917 
38oo 


2io5 


2294 


2482 


2671 


285 9 


3o48 


3236 


3424 


188 


231 


36i2 


3988 


4176 


4363 


455i 


4739 


4926 


5n3 


53oi 


188 


232 


5488 


56 7 5 


5862 


6049 


6236 


6423 


6610 


6796 


6 9 83 


7169 


187 


233 


7356 


7542 


7729 


7 9 i5 


8101 


8287 


8473 


865 9 


8845 


9o3o 


186 


234 


Q2l6 


94oi 


9 58 7 


9772 


99 58 
1806 


•i43 


•328 


•5i3 


•6.98 


•883 


i85 


235 


371068 


1253 


1437 


1622 


3831 


2175 


236o 


2544 


2728 


184 


236 


2912 


3096 


3280 


3464 


3647 


401 5 


4198 


4382 


4565 


184 


237 


4748 


4g32 


5n5 


5298 


5481 


5664 


5846 


6029 


6212 


63g4 


1 83 


238 


65 77 


6759 
858o 


6942 


7124 


73o6 


7488 


7670 


7 852 


8o34 


82J.6 


182 


239 


83 9 8 


8761 


8943 


9124 


93o6 


9487 


9668 


9849 


•«3o 


181 


i 240 


38o2ii 


0392 


0573 


0754 


0934 


1 1 1 5 


1296 


1476 


i656 


i83 7 


181 


241 ! 


2017 


2197 


23 77 


2557 


2737 


2917 


3o 97 


3277 


3456 


3636 


180 


242 i 


38i5 


3995 


4i74 


4353 


4533 


4712 


4891 


5070 


5249 


5428 


S 


243 


56o6 


5 7 85 
7 568 


5964 


6142 


632i 


6499 


6677 


6856 


7034 
8811 


7212 


244 


7 3 9 o 


7746 


7923 


8101 


8279 


8456 


8634 


8989 


178 


245 


9166 


9343 


9520 


9698 


9875 


••5i 


•228 


•4o5 


•582 


• 7 5 9 


i77 


246 


390935 


11 12 


1288 


1464 


1641 


1817 


$? 


2169 


2345 


2521 


176 


247 


2697 


2873 


3o48 


3224 


34oo 


35 7 5 


3926 


4101 


4277 


176 


248 


4452 


4627 


4802 


4977 


5i52 


5326 


55oi 


5676 


585o 


6025 


i 7 5 


249 


6199 


63 7 4 


6548 


6722 


6896 


7071 


7245 


7419 


7 5 9 2 


7766 


174 


250 


397940 


8114 


8287 


8461 


8634 


8808 


8981 


9i54 


9 328 


9501 


i 7 3 


251 


9674 


9847 


••20 


•192 


•365 


•538 


•711 


•883 


io56 


1228 


i 7 3 


252 


401 40 1 


i5 7 3 


1745 


1917 

3635 


2089 


2261 


2433 


26o5 


2777 


294Q 


172 


253 


3i 21 


3292 


3464 


3807 


3 97 8 


4149 


4320 


4492 


4663 


171 


254 


4834 


5oo5 


5n6 


5346 


55i 7 


5688 


5858 


6029 


6199 


6370 


171 


255 


654o 


6710 


6881 


7001 


7221 


7 3oi 
9087 


7061 


773i 


7901 


8070 


170 


256 


8240 


8410 


8579 


8749 


8918 


9 25 7 


9426 


9D95 


9764 


169 


257 


9933 


•102 


•271 


•440 


•609 


•777 


•946 


1114 


1283 


i45i 


;ti 


258 


411620 


1788 


i 9 56 


2124 


2293 


2461 


2629 


2796 


2964 


3i32 


259 


33oo 


3467 


3635 


38o3 


3970 


4i3 7 


43o0 


4472 


4639 


4806 


167 


260 


414973 


5i4o 


5307 


5474 


564i 


58o8 


5974 


61 41 


63o8 


6474 


167 


261 


6641 


6807 


6 97 3 


7i3o 
8798 


73o6 


7472 


7638 


7804 


7970 


8i35 


166 


262 


83oi 


8467 


8633 


8964 


9129 


9 2 9 5 


9460 


9625 


9701 


1 65 


263 


9956 


°I2I 


•286 


•45 1 


•616 


•781 


•945 


1110 


12^5 


1439 


i65 


264 


421604 


I768 


i g 33 


2007 

3 P7 

53 7 i 


2261 


2426 


2090 


2754 


2918 
4555 


3o82 


164 


265 


3246 


3410 


35 7 4 


3901 


4665 


4228 


4392 


4718 


1 64 


265 


4882 


5o45 


5208 


5534 


5697 


586o 


6023 


6186 


634o 

7973 


1 63 


267 


65u 


6674 


6836 


6999 
8621 


7161 


7324 


7486 


7648 


781 1 


162 


268 


8i35 


8297 


845o 


8 7 83 


8944 


9106 


9268 


9429 


9 5 9 i 


162 


269 


97 52 


9914 


••7D 


•236 


•3 9 8 


•55 9 


•720 


•88 1 


1042 


1203 


161 


270 


43 1 364 


i5q5 


i685 


1846 


2007 


2167 


2328 


2488 


2649 


2809 


161 


271 


2969 
456o 
6i63 


3i3o 


3290 


345o 


36io 


3770 


3930 


4.090 
5685 


4249 


4409 


160 


272 


4729 


4888 


5o48 


5207 


5367 


5526 


5844 


6004 


i5 9 


273 


6322 


6481 


6640 


6709 


6957 


7116 
8701 


7275 


7433 


7592 


\U 


274 


7701 


7909 


8067 


8226 


83S4 


8542 


885 9 


9017 


9175 


275 


9333 


9491 


9648 


9806 


9964 


•122 


•279 


•43 7 


•5 9 4 


•752 


1 58 


276 


440909 


1066 


1224 


1 38 1 


1 538 


1695 


1802 


2009 


2166 


2323 


1 5 7 


277 


2480 


2637 


2 79 3 


2o5o 
45i3 


3 1 06 


3263 


34i9 


35 7 6 


3 7 32 


3889 


i5 7 ' 


278 


4o45 


4201 


4357 


4669 


4825 


4981 


5i3 7 


5293 


5449 


1 56 


279 


56o4 


5760 


5915 


6071 


6226 


6382 


6037 


6692 


6848 


7003 


1.55 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 



LOGARITHMS OF NUMBERS. 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


280 
281 
282 
283 
284 
285 
286 
287 
288 
289 


447 i 58 
8706 

450249 
1786 
33i8 
4845 
6366 
7882 
q3 9 2 

460898 


73 13 

8861 
o4o3 
1940 
3471 
4997 
65i8 
8o33 
9543 
1048 


7468 
9015 
o557 
2093 
3624 
5i5o 
6670 
8184 
9694 
1 198 


7623 
9170 

071 1 
2247 

3777 
53o2 
6821 
8336 
9845 
1 348 


7778 
9324 
o865 
2400 
3930 
5454 
6973 
8487 
999 5 

1499 


7 9 33 
9478 
1018 
2553 
4082 
56o6 
7125 
8638 
•146 
1649 


8088 
9 633 
1 172 

2706 
4235 
5758 
7276 
8789 
•296 
H99 


8242 

9787 
i326 
285 9 
438 7 
5910 
7428 
8940 
•447 
1948 


83 9 7 
9941 
U79 

3012 

4540 
6062 

7579 
9091 
•597 

2098 


8552 
••95 
i633 
3i65 
4692 
6214 
77 3i 
9242 
•748 
2248 


1 55 

i54 
1 54 
1 53 
1 53 

152 
l52 

i5i 
i5i 
i5o 


290 
291 
292 
213 
294 
295 
296 
297 
298 
299 
! 


462398 
. 38 9 3 
5383 
6868 
8347 
9822 
471292 
27D6 
4216 
5671 


2548 
4042 
5532 
7016 
8495 
9960 
U38 

2O03 

4362 
58i6 


2697 
4191 
568o 
7164 
8643 
•116 
1 585 
3 049 
45o8 
5962 


2847 
434o 
5829 
7312 
8790 
•263 
1732 
3i 9 5 
4653 
6107 


2997 

4490 
5977 
746o 
8 9 38 
•410 
1878 
334i 

4799 
6252 


3 146 

4639 
6126 
7608 
9085 
•557 

2025 
3487 
4944 

6397 


3296 
4788 
6274 
7756 
9233 
•704 
2171 
3633 
5090 
6542 


3445 
4936 
6423 

9 38o 
•85i 
23i8 
3779 
5235 
6687 


3594 

5o85 
65 7 i 
8o52 
9 52 7 
• 99 8 
2464 
3925 
538 1 
6832 


3744 
5234 
6719 
8200 
9675 
1 U5 
2610 
4071 
5526 
6976 


i5o 

149 

$ 

148 
147 
146 
146 
146 
145 


300 
301 
302 
303 
304 
305 
806 
307 
308 
309 

| 310 
] 311 

312 
313 
314 
815 
316 
317 
318 
319 


477^1 
8566 

480007 
1443 
2874 
43 00 
5721 
7 i38 
855i 
9958 


7266 
871 1 
1 5 1 
i586 
3oi6 
4442 
5863 
7280 
8692 
8 *99 


7411 
8855 
0294 
1729 
3 1 59 
4585 
6oo5 
7421 
8833 
•239 


7555 
8999 
0438 
1872 
33o2 
4727 
6147 
7 563 
8974 
•3 80 


7700 
9143 
o582 
2016 
3445 
4869 
6289 
7704 
9114 

•520 


7844 
9287 
0725 
2159 
358 7 
5on 
643o 
7845 
9255 
•661 

2062 
3458 
485o 
6238 
7621 
8999 
•3 7 4 
1744 
3109 
4471 


7989 
943 1 
0869 

2302 
3730 

5i53 
6572 
7986 
9396 
•801 


8i33 
9 5 7 5 
1012 
2445 
3872 
5295 
6714 
8127 
9 53 7 
•941 


8278 
9719 

1 136 

2588 
401 5 
5437 
6855 
8269 
9677 
1081 


8422 
9 863 
1299 
2731 
4i57 
5579 

6997 
8410 
9818 
1222 


145 

144 
144 
143 
143 
142 
142 
141 
141 
140 


491362 
2760 
4i55 
5544 
6g3o 
83u 
9687 

5oio59 
2427 
3791 


i5o2 
2900 

4294 

5683 
7068 
8448 
9824 
1 196 
2564 
3927 


1642 
3 040 
4433 

5822 

7206 
8586 
9962 
i333 

2700 
4o63 


1782 
3179 
4972 
5960 
7344 
8724 
•• 99 
U70 
283 7 
4199 


1922 
3319 
4in 
6099 

7483 
8862 
•236 
1607 
2073 
4335 


2201 
35 9 7 
4989 
6376 

77^9 
9137 
•5n 
1880 
3246 
4607 


2341 
3 7 3 7 
5i28 
65i5 
7897 
9273 
•648 
2017 
3382 
4743 


2481 
38 7 6 
5267 
6653 
8o35 
94i2 
•785 
2 1 54 
35i8 
48 7 8« 


2621 
4oi5 
5406 
6791 

8n3 
955o 
•922 
2291 
3655 
5oi4 


140 
i3 9 
i3o 
1 3 9 
i38 
i38 
i3 7 
i3 7 
i36 
i36 


820 
321 
322 
823 
324 
325 
326 

! 327 
828 

\ 329 


5o5i5o 
65o5 
7836 
9?o3 

5 10343 
1 883 
32i8 
4548 
58 7 4 
| T9 6 


52S6 
6640 
7991 
9 33 7 
0679 
2017 
335i 
4681 
6006 
7328 


5421 
6776 
8126 

9471 
o8i3 

2l5l 

3i84 
48i3 
6i3g 
7460 


5557 
69 1 1 

8260 
9606 
0947 
2284 
36i 7 
4946 
6271 
7 5 9 2 


5693 
7046 
83 9 5 
9740 
1081 
2418 
375o 
5079 
6403 
7724 


5828 
7181 
853o 
9874 

I2l5 

255i 
3883 
52 1 1 
6535 

7855 


5g64 
73i6 

8664 
oo« 9 

1 349 
2684 
4016 
5344 
6668 
7937 


6099 
745 1 
8799 
•i43 
1482 
2818 
4i49 
5476 
6800 
8119 


6234 
7 586 
8934 
•277 
1616 
2951 
4282 
5609 
6932 

8231 


6370 
7721 
9068 
•4n 
1750 
3o84 
44i4 
5741 
7064 
8382 


1 36 

i35 
i35 
1 34 
i34 
i33 
i33 
1 33 

132 

132 


330 
331 
332 
333 
334 
335 
336 
• 337 
338 
889 


1 5i85u 
9828 
52ii38 
2444 
3 7 46 
5o45 
633 9 
763o 

_ 89n 
53o2oo 


8646 
99 5 9 
1269 
2570 
38 7 6 
5174 
6469 
7709 
9045 
6328 


8777 
••90 
1400 
2705 
4006 
53 04 
65g8 
7888 
9*74 
0456 


8909 
•221 

i53o 
2835 
4i36 
5434 
6727 
8016 
9302 
o584 


9040 9171 
•353 «484 
1661 j 1792 
2966 3096 
4266 4396 
5563 56 9 3 
6856 ! 6 9 85 
8i45 I 8274 
943o ; 9559 
0712 , 0840 


93o3 
•6i5 

I922 1 
3226 
4526 

5822 

71 14 

8402 

9687 

0968 


9434 
*74.5 
2o53 
3356 
•4656 
5 9 5i 
7243 
853 r 
9 8i5 
1096 


9 566 
•876 
2i83 
3486 
4785 
6081 
7372 
8660 
9943 

1223 


9697 
1007 
23 14 
36i6 
4gi5 
6210 
75ot 
8788 

i35i 


i3i 
i3i 
i3i 
i3o 
i3o 
129 
129 

S3 

128 


' N. 





1 


2 


3 


4 5 6 


7 


8 


9 


D. 



UOGARITimx- OF KUMEER8. 



N. 





1 


2. 


3 


4 


5 


6- 


7 


8 


9- 


IX 


f. 


840 


53 1 47 9' 


1607 


# 7 34 


1.862 


1990 


2157 


2245 


2872 


25oo 


2627 


128 




341 


2734 


2882 


3009 


3i36 


3204 


3391 


35 18 


3645 


3772 


8899 


127 




842 


4026 


41 53 


4280 


4407 


4534 


4661 


4787 


4914 


5o4i 


5167 


127 


848 


5 294 


0421 


5547 


56 7 4 


58oo 


0927 


6o53 


6180 


6806 


6.48-2 


126 


Hi 


6558 


' 6685 


68 1 1 


6937 


7068 


7189 


73*5 


7441 


7067 


7698 


I26< 




345 


7819 


7945 


8071. 


8197 


8322 


8448 


85 7 4 


8699 


8825 


8901 


126 




846 


9.07b 


9,202 


9827 


9452 


9P78 


9703 


9829 


9904 


**79 


•204 


125 


i 


347 


54o3 29. 


o455 


o58o- 


0705 


o83o 


0955 


1.080 


1 20 5 


i-33o 


1454 


125 




348 


1679. 


1704 


1829 


1953 


2078 


22o3 


2827 


2452. 


2076 


2701 


125 




349 

350 


2820 


2950 


3074 


3i 99 


3323 


3447 


35 7 i 


36g6- 


8820 


3944 


124, 




344068 


4192 


43 16 


444o 


4064 


4688 


4812 


4986 


5a6o 


5s83 


1.24 




351 


53o7 


543i 


5555 


5678 


58o2 


5 0,2.5 


6049 


6172 


6296 


6419 


124 


; 


8©2 


6043 


6666 


6789 


6918 


7086 


7159 


7282 


74a5 


732-9 


7652' 


1.23- 


: 


353 


7775 


7898 


8021 


8U4 


8267 


838 9 


85i2 


8635 


8758 


8881 


128 




354 


9003 


91 26 


9249 


937 i 


9494 


961b 


9739 


9861 


9984 


•106 


123 




355 


55o2 28 


a35i 0478 


0095 


0717 


0840 


0962 


1084 


1206 


1 3 28 


122 




o-'^ 


1400 


1572 


169.4 


1816 


1938 


2000 


2181 


23o3 


2425 


2047 


122 




857 


2:668 


2790 


291 1 


3o33 


3i55 


3276 


8898 


3519 


364a 


8762 


12i 




358 


3883 


4004 


4126 


4247 


4368 


4489 


4610. 


473-1 


4802 


4973 


121 




859 

360 


5oo4 


521.5 


5336 
6044 


5457 
6664 


5578 
6785 


0099 
6905 


5820 


0940 


6061 


6182 


121 

120 




5563o3 


6423 


7026 


7146 


7267 


7^7 


i 


861 


7 Do 7 


7627 


7748 


7868 


7988 


8108 


8228 


8849 


8469 


8589 


120 




86.2 


8709 


8829 


8948 


9068 


9.1.88 


0808 


9428 


9548 


9667 


97g7 


I20 


; 


363 


9907 


©©26 


•146 


•265 


•385 


•5o4 


•624 


•743 


•863 


•982 


M9 


8M 


56 1 1 01 


1221 


1840 


U59 


1 5 7 8 


1698 


1817 


1936 


2o55 


2174 


119 




365 


2293 


2412 


253i 


265a 


2769 


2887 


3oo6 


3i25 


32-44 


8862 


II9 




8 &6 


348i 


36oo 


3718 


3887 


8955 


4074 


4192 


43 1 1 


4429. 


4548 


M 9 . 




867 


4666 


4784 


4908 


502I 


5i39 


5237 


5376 


5494 


56 1 2 


5730. 


I.18 




368 


5848 


0966 


6084 


6202 


6820 


6487 


6555 


6678 


6791 


6909 


t*& 




361) : 


7026 


7144 


7262 


7^79 


7497 


7014 


7782 


7849 


79°7 


8084 


118 


■ 


370 


568202. 


8819 


8436 


8554 


867 1 


8788 


8905 


9028 


9?.4o 


925 7 


ij 7 




371 


9374 


9491 


9608 


9720. 


9842 


99D9 


••76 


•193 


•3o 9 


•426 


117 




372 


570343 


0660 


0776 


089.3 


1010 


1126 


1243 


i 809 


U76 


1592 


117 




373 


1709 


i825 


194a 


2o58 


2174 


2291 


2407 


2028 


2609 


2755 


si6 




374 


2872 


2988 


3 1 04 


3 2 20 


3386 


3402 


3568 


3684 


38oa 


39,15 


*i6 




875 


4o3i 


4U7 


4263 


• 4379 


4494 


4610 


4726 


48ii 


4957 


5072 


1 16 




376 


5i.88 


53o3 


5419 


5534 


5o5o 


O765 


588a 


5996 


61 ii 


6226 


11.5 




377 


634 J 


6457 


6572 


6687 


6802 


6917 


7o3-2 


V 47 


7262 


7>77 


13,5 




878 


7492 
863o^ 


7607 


7722. 


7886 


79 5j 


8066 


8181 


829O 


8410 


8520 


n5 




379 


8754 


8868 


89,8a 


9097 


9212 


9826 


9441 


9555 


9669 


1,1,4 




880 


579784 


9S98 


• 9} 2 


•126 


®24l 


•355 


•469 


•583 


•697 


•8*1 


«4 




88 i 


580925 


1039 


n53 


1267 


1881 


1490 


1608 


1722 


i836 


195a 


1*4 




3-2 


2068 


21.77 


2291 


2404 


2Jl8 


2681 


2740 


2.858 


2972 


3o85 


ii4 




383 


3199 


33i2 


3426 


3539 


8652 


8765 


8879 


3992 


4io5 


4218 


u3 




384 


433 1 


4444 


4557 


4676 


4783 


4896 


5009 


5l22 


5:?35 


5348 


11.3 




385 


5461 


5574 


5686 


5799 


5a j 2 


6024 


61.37 


625o 


6362 


6475 


n3 




386 


6587 


6700 


,68 1 2 


6923 


7637 


7149 


7262 


1*14 


7486 


7,599 


L12 




387 


7711 


7823 


7935 


8047 


8160 


8272 


8384 


8496 


8608 


8720 


112 




388 


8832 


8944 


9o56 


9167 


9279 


9391 


9003 


961 5 


0726 


9888 


112 




389 


99,00 


••61. 


•i 7 3 


•284 


•396 


•507 


•619 


•730 


•842 


•953 


112 




390 


591065 


1 176 


1287 


1899 


>5io 


1621 


1732 


1848 


5965 


2066 


ni> 




ml 


2177 


2288 


2899 


25lO 


2621 


2732 


2843 


2954 


3o64 


3i 7 5 


ni 




392 


3286 


3397 


35o8 


36 1 8 


3729 


384a 


3950 


4061 


4.1 7 1 


4282 


irv 




393 


4398 


45o3 


4614 


47 2 4 


4834 


4945 


5o55 


5i65 


5276 


5386 


»r.o 




3&4 


5496 


5666 


5717 


5827 


0987 6047 


6157 


6267 


63 7 7 


6487 


FIO 




395 


6597 


6707 


6817 


6927 


7037 7146 


7256 


7366 


7476 


7586 


1 10 




396 


7695 


7805 


7914 


8024 


8 1 34 ! 8243 


8353 


8462 


85 7 2 


8681 


no 




397 


8791 


8900 


9009 


91 19 


9228 j 9887 


9446 


9556 9665 


9774 


j-09 




' 898 


9 883 


999 2 


•101 


•210 »3i9 «428 


•537 


•646 


»755 


•864 


109 




399 


600978 




1082 


1 191 


1299 1 i4o8 1.517 


1625 


1734 


1843 


1951 


109 




N. 


1 


2 


3 j 4 5 


6 


7 


8 


9 


D. 





LOGARITHMS OF NUMBERS. 



, 





1 


2 


3 


4 


5 


6 


7 


8 


9 


H 


400 


602060 


2169 


2277 


2386 


2494 


26o3 


2711 


2819 


2928 


3o36 


108 


401 


3 144 


3253 


336 1 


3469 


3577 


3686 


3794 


3902 


4010 


4118 


)o8 


402 


4226 


4334 


4442 


455o 


4658 


4766 


4874 


4982 


5089 


5197 


108 


403 


53o5 


54i3 


552i 


5628 


5736 


5844 


3931 


6059 


6160 


6274 


108 


404 


63 si 


6489 


6596 


6704 


681 1 


6919 


7026 


7i33 


7241 


7348 


107 j 
10 7 ji 


405 


7455 


7062 


7669 


7777 


7884 


7991 


8098 


8203 


83i2 


8419 


40(5 


8026 


8633 


8740 


8847 


8 9 54 


9061 


9167 


9274 


938i 


9488 


107 
107 


407 


9394 


9701 


9808 


9914 


••21 


•128 


•234 


•341 


•447 


•554 


408 


610660 


0767 


0873 


0979 


1086 


1192 


1298 


i4o5 


i5u 


1617 


106 t 


409 


1723 


1829 


1936 


2042 


2148 


2254 


236o 


2466 


2572 


2678 


106 | 


410 


612784 


2890 


2996 


3l02 


32o7 


33i3 


3419 


3525 


363o 


3736 


106 1 


411 


3842 


3947 


4o53 


41 Sg 


4264 


4370 


4475 


458 1 


4686 


4792 


106 jj 


412 


4897 


5oo3 


5io8 


52i3 


5319 


5424 


5529 


5634 


3740 


5845 


io5 I 


413 


5o3o 


6o55 


6160 


6265 


6370 


6476 


658i 


6686 


6790 


6895 


io5 | 


414 


7000 


7io5 


7210 


73i5 


7420 


7525 


7629 


7734 


7889 


7943 


io5 jj 


415 


8048 


81 53 


8207 


8362 


8466 


8571 


8676 


8780 


8884 


8989 


io5 | 


416 


9093 


9198 


9302 


9406 


9011 


9615 


9719 


9824 


9928 


••32 


104 (j 


417 


620136 


0240 


o344 


0448 


o552 


o656 


0760 


0864 


0968 


1072 


104 1 


418 


1176 


1280 


1384 


j 488 


l5 9 2 


1695 


'799 


1903 


2007 


2110 


104 1 


419 
420 


2214 


23i8 


2421 


2525 


2628 


2732 


2835 


2 9 3 9 


3o42 


3i46 


104 J 


623249 


3353 


3456 


3559 


3663 


3 7 66 


386 9 


3973 


4076 


4.179 


io3 I 


421 


4282 


4385 


4488 


4591 


4695 


4798 


4901 


5oo4 


5107 


5210 


io3 8 


422 


53i2 


54i 5 


55i8 


5621 


5724 


5827 


3929 


6o32 


6i35 


6238 


io3 I 


423 


634o 


6443 


6546 


6648 


6751 


6853 


6956 


7o58 


7161 


7263 


io3 j 


424 


7366 


7408 


7 5 7 i 


7673 


7775 


7878 


7980 


8082 


8i85 


8287 


102 I 


425 


838 9 


8491 


85g3 


8695 


S797 


8900 


9002 


9104 


9206 


9308 


102 I 


426 


9410 


9012 


9613 


97 i5 


9817 


9919 


••21 


•123 


•224 


•326 


102 1 


427 


630428 


• o53o 


o63i 


0733 


o835 


0936 


io38 


1 139 


1241 


1342 


102 1 


428 


1444 


i545 


1647 
2660 


1748 


1849 


1951 
2963 


2052 

3064 


2 1 53 

3i65 


2255 


2356 

336 7 


101 I 


429 


2437 


2559 


2761 


2862 


3266 


101 | 


1 430 


633468 


3569 


3670 


3771 


38 7 2 


3973 


4074 


'4175 


4276 


4376 


100 


431 


4477 


4578 


4679 


4779 


4880 


49B1 


5o8i 


5i82 


5283 


5383 


100 


| 432 


5484 


5584 


5685 


5785 


5886 


598b 


0087 


6187 


6287 


6388 


100 


j 433 


6488 


6588 


6688 


6789 


6889 


6989 


7089 


7189 


7290 


7390 


100 


434 


7490 


7 5 9 o 


7690 


7790 


7890 


7990 


8090 


8190 


8290 


838g 


99 


435 


848 9 


8589 


8689 


8789 


8888 


8968 


9088 


9188 


9287 


93^7 


99 ; 


436 


9486 


9586 


9686 


9780 


9 885 


9984 


••84 


»i83 


•283 


•382 


99 ! 


437 


640481 


o58i 


0680 


°779 


0879 


0978 


1077 


1177 


1276 


i375 


99 


438 


1474 


1573 


1672 


1771 


1871 


1970 


2069 


2168 


2267 


2366 


99 


439 


2460 


2563 


2662 


2761 


2860 


2939 


3o58 


3i56 
4U3 


3255 
4242 


3354 
434o 


99 j 

9 8 J 


440 


643453 


355i 


365o 


3749 


3847 


3946 


4o44 


441 


4439 


4537 


4636 


4734 


4832 


493i 


5029 


5127 


0226 


5324 


98 : 


442 


5422 


552i 


0619 


5717 


58 1 5 


5913 


601 1 


61 10 


6208 


63oo 


98 ! 


443 


6404 


65o2 


6600 


6698 


6796 


6894 


6992 


7089 


7187 


7285 


98 | 


444 


7383 


748i 


7579 


7676 


7774 


7872 


7969 


8067 


8i65 


8262 


98 


445 


836o 


8458 


8535 


8653 


875o 


8848 


8945 


9043 


9 1 40 


9237. 


97 1 


446 


9335 


9432 


9380 


9627 


9724 


9821 


9919 


••16 ' 


•n3- 


•210 


97 i 


447 


65o3o8 


040 5 


0302 


0599 


0696 


0793 


0890 


0987 


1084 


i.i8i 


97 


448 


1278 


1375 


1472 


1 569 


1 066 


1762 


1859 


1956 


2o53 


2i5o 


97 1 


449 


2246 


2343 


2440 


253o 


2633 


2730 


2826 


2923 
3888 


3019 
3984 


3u6 
40S0 


97 f 
96 f 


1 450 


6532i3 


3309 


34o5 


35o2 


3598 


3695 


3791 


451 


im 


4273 


4369 


4465 


4562 


4658 


4734 


485o 


4946 


5o42 


96 [■ 


452 


oi 38 


5235 


533i 


5427 


5523 


3619 


57*5 


58io 


5906 


6002 


9 6 I 


453 


6098 


6194 


6290 


6386 6482 


6577 


66 7 3 


6769 


6864 


6960 


96 


454 


7o56 


7162 


7247 


7343 


7438 


7534 


7629 


7725 


7S20 


7916 


96 


455 


801 1 


8107 


8202 


8298 


83 9 3 


8488 


8584 


8679 


8774 


8870 


9 5 


456 


8965 


9060 


91 55 


9200 


9346 


9441 


9536 


963 1 


9726 


9821 


95 


457 


9916 


••11 


•106 


•201 


•296 


•391 


•486 


•58 1 


®676 


•771 


9 5 


458 


66o865 


0960 


io55 


ii5o 


1245 


1339 


1434 


i52-9 


r623 


1718 


95 


459 
N. 


i8j3 


1907 


2002 


2090 ! 2191 


2 280 


238o 


2473 


2569 


2663 


95 





1 


2 


S 4 


5 


6 


7 


8 


9 . 


D. 



LOGARITHMS OF NUMBERS. 



N. 





1 


2 


8 


4 


5 


6 


7 


8 


9 


D. 


460 


662758 


2852 


2947 


3o4i 


3i35 


323o 


3324 


3418 


35i2 


3607 
4548 


94 


461 


3701 


3795 


388 9 


3 9 83 


4078 


4172 


4266 


436o 


4454 


94 


462 


4642 


4736 


483o 


4924 


5oi8 


5ll2 


5206 


5299 


53 9 3 


5487 


94 


463 


558i 


5675 


5769 


586a 


5 9 56 


6o5o 


6i43 


6237 


633 1 


6424 


94 


464 


65i8 


6612 


670D 


6799 


6892 


6986 


7079 
8oi3 


7173 


7266 
8199 


7360 
8293 


94 


465 


7453 


7546 
8479 


7640 
85 7 2 


77 33 


7826 


7920 


8106 


93 


466 


8386 


8665 


8 7 5 9 


8852 


8945 


9 o38 


9i3i 


9224 


93 


467 


9 3i 7 


9410 


95o3 


9596 


9689 


9782 


9875 


9967 
0895 


••60 


•i53 


9 3 


468 


670246 


o339 


o43 1 


o524 


0617 


0710 


0802 


0988 


1080 


9 3 


469 


1173 


1266 


i358 


i45i 


1043 


1 636 


1728 


1821 


J913 


2005 


93 


470 


672098 


2190 


2283 


23 7 5 


2467 


256o 


2652 


2744 


2836 


2929 


92 


471 


3021 


3n3 


32o5 


3297 


3390 


3482 


3574 


3666 


3 7 58 


385o 


92 


472 


3942 


4o34 


4126 


4218 


43 10 


4402 


4494 


4586 


4677 


4769 


92 


473 


4861 


4953 
5870 


5o45 


5i37 


5228 


5320 


5412 


55o3 


5595 


568 7 


92 


474 


5778 


5962 


6o53 


6i45 


6236 


6328 


6419 


65u 


6602 


92 


475 


6694 


6 7 85 


6876 


6968 


7 o5 9 


7i5i 


7242 


7333 


7424 


7516 


91 


476 


7607 


7698 


7789 
8700 


7881 
8791 


7972 
8882 


8o63 


8i54 


8245 


8336 


91 


477 


85i8 


8609 


8 97 3 


9064 


9165 


9246 


9337 


91 


478 


9428 


9 5l 9 


9610 


9700 


9791 


9882 


997 3 


••63 


•i54 


•245 


9 1 


479 


68o336 


0426 


0517 


0607 


0698 


0789 


0879 


0970 


1060 


ii5i 


91 


480 


681241 


i332 


1422 


i5i3 


i6o3 


1693 


1784 


1874 


1964 


2o55 


90 


481 


2i45 


2235 


2320 


2416 


25o6 


2596 


2686 


2777 


2867 


2 9 5 7 


90 


482 


3o47 


3i3 7 


3227 


33i7 


3407 


3497 


358 7 


36 77 


3767 


3837 


90 


483 


3947 


4037 


4127 


4217 


4307 


4396 


4486 


4576 


4666 


4 7 56 


90 


484 


4845 


4935 


5o25 


5n4 


5204 


5294 


5383 


5473 


5563 


5652 


90 


485 


5742 


583 1 


5921 


6010 


6100 


6189 


6279 


6368 


6458 


6547 


£ 9 


486 


6636 


6726 


68i5 


6904 


6994 


7083 


7172 


7261 


735i 


744o 


S» 


487 


7529, 


7618 


7707 


7796 


7886 
8776 


7975 


8064 


8i53 


8242 


833 1 


^ 


488 


8420 


8509 


85 9 8 


8687 


8865 


8953 


9042 


9i3i 


9220 


8 o 9 


489 


9309 


9398 


9486 


9575 


9664 


97 53 


9841 


9930 


••19 


•107 


89 


490 


690196 


0285 


0373 


0462 


o55o 


0639 


0728 


0816 


0905 


0993 


ll 


491 


1081 


1170 


1258 


i347 


1435 


i524 


1612 


1700 


1789 


1877 
2 7 5 9 


88 


492 


1965 


2o53 


2142 


2230 


23 18 


2406 


24.94 


2583 


2671 


88 


493 


2847 


2935 


3o23 


3i 1 1 


3 199 


3287 


3375 


3463 


355i 


3639 


88 


494 


37.27 


38i5 


3903 


3991 


4078 


4166 


4254 


4342 


443 


45i7 


88 


495 


460 5 


4693 


4781 


4868 


4936 


5o44 


5i3i 


5219 


5307 


53 9 4 


88 


496 


5482 


5569 


56j7 


5744 


5832 


5919 


6007 


6094 


6182 


6269 


V 


497 


6356 


6444 


653 1 


6618 


6706 


6 79 3 


6880 


6968 


7o55 


7142 


I 1 


498 


7229 


73 \i 


7404 


7491 


7 5 7 8 


7665 


77 52 

8622 


7839 


7926 


8014 


87 


499 


8101 


8188 


8273 


8362 


8449 


8535 


8709 


8796 


8883 


87 


500 


698970 


9 o5 7 


9'44 


923i 


9317 


9404 


9491 


9 5 7 8 


0664 


97 5i 


87 


501 


9 838 


9924 


es,, 


•® 9 S 


•184 


•271 


•3o8 


•444 


6 53 1 


•617 


87 


502 


700704 


0790 


0877 


0963 


io5o 


u36 


1222 


1 309 


1395 


1482 


86 


503 


1 568 


j 654 


1741 


1827 


1913 


1999 


2086 


2172 


2258 


2344 


86 


504 


243i 


2317 


26o3 


2689 


2 77 5 


2861 


2947 


3o33 


3n 9 


32o5 


86 


50.") 


3291 
4i5i 


33 77 


3463 


3549 


3635 


3 7 2i 


3807 


38 9 3 


3979 


4o65 


86 


506 


4236 


432 2 


4408 


4494 


4079 


4665 


473i 


483 7 


4922 


86 


507 


5oo8 


5094 


5179 


5265 


535o 


5436 


5522 


5607 


0693 


5 77 8 


86 


508 


5864 


0949 


6o35 


6120 


6206 


6291 


63 7 6 


6462 


6547 


6632 


85 


509 


6718 


68o3 


6888 


6974 


7 o5 9 


7U4 


7229 


73i5 


7400 


7 4S5 


85 


510 


707570 


7655 


774o 


7826 


7911 


7996 


8081 


.8166 


825i 


8336 


85 


511 


8421 


85o6 


8091 


8676 


8761 


8846 


8 9 3i 


9015 


9100 


9185 


85 


512 


9270 


9 355 


944o 


9524 


9609 


9694 


9779 


9863 


9948 


••33 


85 


513 


710117 


0202 


0287 


0371 


0456 


o54o 


0620 


0710 


0794 


0879 


f 


514 


0963 


1048 


Il32 


1217 


i3oi 


i385 


1470 


1 554 


i63g 


1723 


84 


515 


1807 


1892 
2734 


1976 


2060 


2144 


2229 


23i3 


2397 


2481 


2566 


84 


516 


265o 


2818 


2902 


2986 


3070 


3i54 


3238 


3323 


3407 


84 


517 


3491 


3575 


365g 


3742 


3826 


3910 


3994 


4078 


4162 


4246 


84 


518 


433o 


44i4 


4497 
5335 


458 1 


4665 


4749 


4833 


4916 


3COO 


5o84 


Si 


519 


5i6 7 



5201 


5418 


55o2 


5-586 


566 9 


5 7 53 


5836 


6920 


84 


N. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 



LOGAEITHMS OF NUMBEKS. 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


520 


7i6oo3 


6087 


6170 


6254 


6337 


6421 


65o4 


6588 


6671 


6754 

758 7 


83 


521 


6838 


6921 


7004 


7088 


7171 


7254 


7338 


7421 


75o4 


83 


522 


7671 


8585 


8668 


7920 


8oo3 


8086 


8169 


8253 


8336 


8419 
9248 


83 


523 


85o2 


8751 


8834 


8917 


9000 


9083 


9165 


83 


524 


933 1 


9414 


9497 


9080 


9 663 


974D 


9828 


991 1 


9994 
0821 


eo 77 


83 


525 


720159 


0242 


o325 


0407 


0490 


0573 


o655 


0738 
1 563 


0903 


83 


526 


0986 
1811 


1068 


ii5i 


1233 


i3i6 


1398 


1481 


1646 


1728 


82 


527 


i8 9 3 


1975 


2o58 


2140 


2222 


23o5 


238 7 


2469 


2552 


82 


528 


2634 


2716 
3538 


2798 


2881 


2963 


3o45 


3127 


3209 


3291 


3374 


82 


529 
530 


3456 


3620 


3702 


3784 


3866 


3948 


4o3o 


4112 


4194 


82 


724276 


4358 


444o 


4522 


4604 


4685 


4767 


4849 


493i 


5oi3 


82 


531 


&095 


5i 7 6 


5258 


534o 


5422 


55o3 


5585 


5667 


5748 


583o 


82 


532 


5912 


5993 
6809 
7623 


6075 


6i56 


6238 


6320 


6401 


6483 


6564 


6646 


82 


533 


6727 


6890 


6972 


7o53 


7i34 


7216 


7297 


7379 


746o 


81 


534 


7541 


7704 


77 85 


7866 


7948 


8029 


8110 


8191 


8273 


81 


535 


8354 


8435 


85i6 


85 9 7 


8678 


8 7 5 9 


8841 


8922 


9003 


9084 


81 


536 


9165 


9246 


9 32 7 


9408 


9489 
•298 


9 5 7 o 


965i 


9732 


9 8i3 


9893 


81 


537 


9974 


••55 


•i36 


•217 


•378 


•45 9 


•54o 


•621 


•702 


81 


538 


730782 


o863 


0944 


1024 


no5 


1 186 


1266 


1347 


1428 


i5o8 


81 


539 


i58 9 


1669 


1750 


i83o 


1911 


1991 


2072 


2l52 


2233 


23i3 


81 


540 


732394 


2474 


2555 


2635 


2715 


2796 


2876 


2956 


3o37 


3i 17 


80 


541 


3i 97 


3278 


3358 


3438 


35i8 


35 9 8 


3679 


3759 


383 9 


3919 


80 


542 


3999 


4079 


4160 


4240 


4320 


44oo 


4480 


456o 


4640 


4720 


80 


543 


4800 


4880 


4960 


5o4o 


5l20 


5200 


5279 


5359 


5439 


5519 


80 


544 


55 99 


5679 


5 7 5 9 


5838 


5918 


5998 


6078 


6i5 7 


623 7 


63i 7 


80 


545 


6397 


6476 


6556 


6635 


6 7 i5 


6795 


6874 


6 9 54 


7034 


71 13 


80 


546 


7193 


7272 


7352 


743i 


701 1 


7590 


7670 


7749 


7829 


7908 


79 


547 


7987 


8067 


8146 


8225 


83o5 


8384 


8463 


8543 


8622 


8701 


79 


548 


8781 


8860 


8 9 3 9 


9018 


9097 
9889 


9177 


9 256 


9335 


94U 


94q3 


79 


549 


9 5 7 2 


9601 


9781 


9810 


9968 


••47 


•126 


°2o5 


•284 


79 


550 


74o363 


0442 


0521 


0600 


0678 


075-7 
1 546 


o836 


0915 


0994 


1073 


79 


551 


Il52 


1230 


1309 


i388 


1467 


1624 


1703 


1782 


i860 


79 


552 


i 9 3 9 


2018 


2096 


2173 


2254 


2332 


241 1 


*248 9 
3275 


2568 


2647 


79 


553 


272D 


2804 


2882 


2961 


3o39 


3u8 


3196 


3353 


343 1 


78 


554 


35io 


3588 


3667 


3i45 


3823 


3902 


3980 


4o58 


4i36 


42i5 


78 


555 


42 9 3 


43 7 i 


4449 


4528 


4606 


4684 


4762 


4840 


4919 


4997 


78 


556 


5075 


5i53 


523i 


5309 


5387 


5465 


5543 


D621 


0699 


5 777 


78 


557 


5855 


5 9 33 


601 1 


6089 


6167 


6245 


6323 


6401 


6479 


6556 ' 


78 


558 


6634 


6712 


6790 


6868 


6945 


7023 


7101 


7179 
79D5 


7206 


7334 
8110 


78 


559 


7412 


7489 


7067 


7645 


7722 


7800 


7878 


8o33 


78 


| 560 


748188 


8266 


8343 


8421 


8498 


85 7 6 


8653 


8 7 3i 


8808 


8885 


77 


561 


8 9 63 


9040 


9118 


9 i 9 5 


9272 


935o 


9427 


9504 


9582 


9659 


77 


562 


9736 


0814 


9891 


9968. 


«©45 


•i23 


•200 


•277 


•354 


•43 1 


77 


563 


75o5o8 


o586 


o663 


0740 


0817 


0894 


0971 


1048 


I I 25 


1202 


77 


564 


1279 


i356 


1433 


i5io 


1687 


1664 


i74i 


1818 


1895 


1972 


77 


565 


2048 


2125 


2202 


2279 


2356 


2433 


2509 


2586 


2663 


2740 


77 


566 


2816 


2893 


2970 


3o47 


3i23 


3200 


3277 


3353 


343o 


35o6 


11 


567 


3583 


366o 


3736 


38i3 


388g 


3966 


4042 


4119 


4195 


4272 


77 1 


568 


4348 


4425 


45oi 


4578 


4654 


4730 


4807 


4883 


4960 


5o36 


7 6 


569 


5lI2 


5i8 9 


5265 


5341 


5417 


5494 


5570 


5646 


5722 


5799 


76 


570 


7558 7 5 


5901 


6027 


6io3 


6180 


6256 


6332 


6408 


6484 


656o 


76 


571 


6636 


6712 


6788 


6864 


6940 


7016 


7092 


7168 


7244 


7320 


76 


572 


73 9 6 


7472 


7548 


7624 


7700 


7775 


785i 


7927 


8oo3 


8079 


76 


573 


8i55 


823o 


83o6 


8382 


8458 


8533 


8609 


8685 


8761 


8836 


76 


574 


8912 


8988 


9063 


9 i3 9 


9214 


9290 


9 366 


9441 


9517 


9 5 9 2 


76 


575 


9668 


9743 


9819 
0073 


9894 


9970 


••45 


°12I 


•196 


© 2 -7 2 


6 34 7 


7 5 


576 


760422 


0498 


0649 


0724 


0799 


0875 


09D0 


1025 


1 101 


7^ 


577 


1 176 


I25l 


i326" 


1402 


1477 


i552 


1627 


1702 


1778 


1 853 


7 5 


578 


1928 


2003 


2078 


2 1 53 


2228 


23o3 


23 7 8 


2453 


2029 


2604 


t 


579 


2679 


2754 


2829 


2904 


2978 


3o53 


3i28 


32o3 


3278 


3353 


N. 


1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 



10 



LOGARITHMS OF NUMBEES. 




LOGARITHMS OF NUMBERS. 



11 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


640 


806180 


6248 


63i6 


6384 


645i 


65i9 


658 7 


6655 


6723 


6790 


68 


641 


6858 


6926 


6994 


7061 


7129 


7197 


7264 


7 332 


74oo 


7467 


68 


642 


7535 


7603 


7670 


77 38 


7806 


7873 


794i 


8008 


8076 


8i43 


68 


648 


8211 


8279 


8346 


84i4 


8481 


8549 


8616 


8684 


8751 


8818 


67 


644 


8886 


8953 


9021 


9088 


9i56 


9223 


9290 


9358 


9425 


9492 


67 


645 


9560 


9627 


9694 


9762 


9829 


9896 


9964 


•03 1 


•® 9 8 


•i65 


67 


646 


8io233 


o3oo 


o367 


0434 


o5oi 


o56>9 


o636 


0703 


0770 


0837 


67 


647 


0904 
1673 


0971 


1039 


1 106 


1173 


1240 


1307 


1374 


I44i 


i5o8 


67 


648 


1642 


1709 


1776 


i843 


1910 


1977 


2044 


21 1 1 


2178 


67 


649 


2245 


23l2 


2379 


2445 


25l2 


25 79 


2646 


2713 


2780 


2847 


67 
67 


650 


812913 


2980 


3o47 


3n4 


3i8i 


3247 


33i4 


338i 


3448 


35i4 


651 


358i 


3648 


37i4 


3781 


3848 


3 9 i4 


3981 


4048 


4ii4 


4181 


67 


652 


4248 


43i4 


438i 


4447 


45i4 


458i 


4647 


4714 


4780 


4847 


67 


653 


49i3 


4980 


5o46 


5n3 


5179 


5246 


53i2 


5378 


5445 


55n 


66 


654 


5578 


5644 


5 7 m 


5777 


5843 


5910 


5976 


6042 


6109 


6175 


66 


655 


6241 


63o8 


63 7 4 


6440 


65o6 


6573 


663 9 


6700 


6771 


6838 


66 


856 


6904 


6970 


7036 


7102 


7169 


7235 


73oi 


736 7 


7433 


7499 


66 


657 


7565 
8226 


763 1 


7698 


7764 


783o 


7896 


7962 


8028 


8094 
8 7 54 


8160 


66 


658 


8292 


8338 


8424 


8490 


8556 


8622 


8688 


8820 


66 


659 


8885 


8951 


9017 


9083 


9149 


9215 


9281 


9346 


94i2 


9478 


66 


660 


819544 


9610 


9676 


9741 


9807 


9873 


9939 


•••4 


••70 


•i36 


66 


661 


820201 


0267 


o333 


0399 
io55 


0464 


o53o 


0590 


0661 


0727 


0792 


66 


662 


o858 


0924 


0989 


1 1 20 


1 186 


I25l 


1317 


i382 


1448 


66 


663 


i5i4 


1579 


i645 


1710 


1775 


1 841 


1906 


1972 


2037 


2103 


65 


664 


2168 


2233 


2299 


2364 


243o 


2495 


256o 


2626 


2691 


2756 


65 


665 


2822 


2887 


2952 


3oi8 


3o83 


3i48 


32i3 


3279 


3344 


3409 


65 


666 


3474 


353 9 


36o5 


3670 


3 7 35 


38oo 


3865 


3930 


3996 


4061 


65 


667 


4126 


4191 


4256 


4321 


4386 


445 1 


45i6 


458i 


4646 


4711 


65 


668 


4776 


4841 


4906 
5556 


4971 


5o36 


5ioi 


5i66 


523i 


5296 


536i 


65 


669 


5426 


5491 


562i 


5686 


575i 


58i5 


588o 


5945 


6010 


65 


670 


826075 


6140 


6204 


6269 


6334 


6399 


6464 


6528 


65 9 3 


6658 


65 


671 


6723 


6787 


6852 


6917 
7563 


6981 


7046 


7111 


7175 


7240 


73o5 


65 


672 


736 9 


7434 


7499 


7628 


7692 


7757 


7821 


7886 


7 9 5 1 


65 


673 


8oi5 


8080 


8144 


8209 


8273 


8338 


8402 


8467 


853 1 


85 9 5 


64 


674 


8660 


8724 


8789 


8853 


8918 
9361 


8982 


9046 


91 1 1 


9175 


9 23 9 


64 


675 


93o4 


9 368 


9432 


9497 


9625 


9690 


9754 


9818 


9882 


64 


676 


9947 


••11 


••75 


•i3 9 


•204 


•268 


•332 


•3 9 6 


•460 


•525 


64 


677 


83o589 


o653 


0717 


0781 


o845 


0909 


0973 


10J7 


1102 


1 166 


. 64 


678 


1230 


1294 


i358 


1422 


i486 


i55o 


1614 


1678 


1742 


1806 


64 


679 


1870 


1934 


1998 


2062 


2126 


2189 


2253 


2317 


238i 


2445 


64 


680 


832509 


25 7 3 


2637 


2700 


2764 


2828 


2892 
353o 


2956 


3020 


3o83 


64 


6S1 


3i47 


321 1 


3275 


3338 


3402 


3466 


3593 
423o 


3657 


3 7 2i 


64 


682 


3 7 84 


3848 


3912 


3975 


4039 


4io3 


4166 


4294 


4357 


64 


683 


442i 


4484 


4548 


461 1 


4675 


4739 


4802 


4866 


4929 


4993 


64 


684 


5o56 


5l20 


5i83 


5247 


53 10 


53 7 3 


5437 


55oo 


5564 


D627 


63 1 


685 


5691 


5754 


58i 7 


588i 


5944 


6007 


6071 


6i34 


6197 


6261 


63 


686 


6324 


638 7 


645 1 


65i4 


65 7 7 


6641 


6704 


6767 


683o 


6894 


63 


687 


6 9 5 7 


7020 


7083 


7146 


7210 


7273 


7 336 


7399 


7462 


7525 


63 f 


688 


7 588 
8219 


7652 


7 7 i5 
8345 


7778 


784i 
8471 


7904 


7967 


8o3o 


8o 9 3 


8i56 


63 


689 


8282 


8408 


8534 


85 9 7 


8660 


8723 


8786 


63 


690 


838849 


8912 


8 97 5 


9038 


9101 


9164 


9227 


9289 


9352 


941 5 


63 


691 


9478 


9541 


9604 


9667 


9729 


979 2 


9 855 


9918 


9981 


••43 


63 


692 


840106 


0169 


0232 


0294 


o357 


0420 


0482 


o545 


0608 


0671 


63 


693 


0733 


0796 


0809 


0921 


0984 


1046 


1 109 


1172 


1234 


1297 


63 


694 


1359 


1422 


1485 


1 547 


1610 


1672 


1735 


1797 


i860 


1922 


63 


695 


1985 


2047 


2110 


2172 


2235 


2297 


236o 


2422 


24S4 


2547 


62 f 


696 


2609 


2672 


2734 


2796 


2859 


2921 


2 9 83 


3o46 


3io8 


3170 


62 


697 


3233 


329D 


3357 


3420 


3482 


3544 


36o6 


366 9 


3 7 3i 


3793 


62 | 


69S 


3855 


3gi8 


3 9 8o 


4042 


4104 


4166 


4229 


4291 


4353 


441 5 


62 


699 


4477 


4539 


4601 


4664 


4726 


4788 


485o 


4912 


4974 


5o36 


62 J 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. [ 



12 



LOGARITHMS OF NUMBERS. 



N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 


700 
701 
702 
703 
704 
705 
706 
707 
708 
709 


845oo8 
5 7 i8 
6337 
6o55 

8189 
88o5 
9419 
85oo33 
0646 


5i6o 
5780 
6399 
7017 
7634 
825i 
8866 
9481 
0095 
0707 


5222 

5842 
6461 

7696 
83i2 
8928 
9542 
oi56 
0769 


5284 
5904 
6523 
7141 

77 58 
83 7 4 
8989 
9604 
0217 
o83o 


5346 
5 9 66 
6585 
7202 
7819 
8435 
9o5i 
9 665 
0279 
0891 


54o8 
6028 
6646 
7264 
7881 

8497 
9112 
9726 
o34o 
0952 


5470 
6090 
6708 
7326 

8?5 9 
9H4 
9788 
0401 
1014 


5532 
6i5i 

6770 
7388 
8004 
8620 
9235 

9849 
0462 
1075 


5594 
62i3 
6832 
7449 
8066 
8682 

9297 
991 1 
o524 
n36 


5656 
6275 
6894 
n5n 
8128 
8743 
9 358 
9972 
o585 
1 197 


62 
62 
62 
62 
62 
62 
61 
61 
61 
61 


710 
711 
712 
718 
714 
715 
716 
717 
718 
719 


85i258 
1870 
2480 
3090 
36 9 8 
43o6 
4qi3 
5019 
6124 
6729 


l320 

iq3i 
2641 
3i5o 
3759 
436 7 

5?8o 
6i85 
6789 


i38i 
1992 
2602 

321 1 

3820 
4428 
5o34 
564o 
6245 
685o 


1442 
2o53 
2663 
3272 
388i 
4488 
6095 
5701 
63o6 
6910 


i5o3 
2114 

2724 
3333 
3g4i 
4549 
5i56 
5761 
6366 
6970 


1 564 
2175 
2785 
33 9 4 
4002 
4610 
52i6 

5822 

6427 
703 1 


1625 

2236 

2846 
3455 

4o63 
4670 
5277 
5882 
6487 
7091 


1686 
2297 

'4124 
473i 
533 7 
5943 
6548 
7i5z 


1747 
2358 
2968 

4i85 

4792 
53 9 8 
6oo3 
6608 
7212 


1809 
2419 
3029 
3637 
4245 
4852 
5459 
6064 
6668 
7272 


61 
61 
61 
61 
6i 
61 
61 
61 
60 
60 


720 
721 
722 
723 
724 
725 
726 
727 
728 
729 


85 7 332 
7935 
853 7 
9i38 
9739 

86o338 
0937 
1 534 

2l3l 

2728 


73 9 3 

$ 

9198 

o3g8 
0996 
1594 
2191 

2787 


7453 
8o56 
865 7 
9258 
9 85 9 
0458 
io56 
i654 

225l 

2847 


7 5i3 
8116 
8718 
93i8 
9918 
o5i8 
1116 
1714 

23lO 

2906 


7574 
8176 
8778 
9379 
9978 
0578 
1176 
1773 
2370 
2966 


7634 

8236 
8838 
943o 
••38 
0637 
1236 
1833 
243o 
3o25 


7694 
8297 
8898 

9499 
••98 
0697 
1295 
i8o3 

2489 
3o85 


7755 
8357 
8o58 
9559 
•i58 
0757 
i355 
1952 
2549 
3i44 


7 8i5 

8417 
9018 
9619 
•218 
0817 
i4i5 
2012 
2608 
3204 


7875 
8477 
9078 
9679 
•278 
0877 
1475 
2072 
2668 
3263 


60 
60 
60 
60 
60 
60 
60 
60 
60 
60 


730 
731 
732 
733 
734 
735 
736 
737 
738 
739 


863323 
3qi 7 
45u 
5io4 
56 9 6 
6287 
6878 
7467 
8o56 
8644 


3382 
3977 
4570 
5i63 
5755 
6346 
6 9 3 7 
7526 
8n5 
8703 


3442 
4o36 
463o 

5222 

58i4 
64o5 
6996 
7 585 
8174 
8762 


35oi 
4096 
4689 
5282 
58 7 4 
6465 
7o55 
7644 
8233 
8821 


356i 
4i55 
4748 
534i 
5o33 
6524 
7114 
7703 
8292 
8879 


3620 
4214 
4808 
5400 
5 99 2 
6583 
7173 
7762 
835o 
8 9 38 


368o 
4274 
4867 
5459 
6o5i 
6642 
7232 
7821 
8409 
8997 


3739 
4333 
4926 
55i9 
6110 
6701 
7291 
7880 
8468 
9056 


3799 
43o2 
4g85 
55 7 8 
6169 
6760 
735o 

79 3 9 
852 7 

91U 


3858 
4452 
5o45 
5637 
6228 
6819 
7409 
7998 
8586 
9173 


5 9 

t 9 
59 

59 


740 
741 
742 
743 
744 
745 
746 
747 
748 
749 


869232 

9818 

870404 

$ 

2i56 
2 7 3 9 
332i 
3902 
4482 


9290 

9877 
0462 
1047 
i63i 

22l5 

2797 
3379 
3960 
4540 


9349 
9935 
0621 
1 106 
1690 
2273 
2855 
3437 
4018 
45 9 8 


9408 
9994 
o5 79 
1 164 
1748 
233i 
2913 
3495 
4076 
4656 


9466 
••53 
o638 

1223 

1806 
2389 
2972 

3553 
4i34 

47U 


9525 
•in 

0696 
1281 
1 865 
2448 
3o3o 
36n 
4192 
4772 


9 584 
•170 
0755 
i33g 

1023 

25o6 
3o88 
366 9 
425o 
483o 


9642 
•228 
o8i3 
1 3 9 8 
1981 
2564 
3i46 
3727 
43 08 
4888 


9701 
•287 
0872 
1456 
2040 
2622 
3204 
3785 
4366 
4945 


9760 
•345 
0930 
i5i5 
2098 
2681 
3262 
3844 
4424 
5oo3 


< 9 
59 

58 
58 
58 
58 
58 
58 
58 
58 


750 
751 
752 
753 
754 
755 
756 
757 
758 
759 


875061 
564o 
6218 
6795 
7371 
7947 

8522 

9096 

9669 
880242 


5i 19 

56 9 8 
6276 
6853 
7429 
8004 
85 79 
9i53 
9726 
0299 


5177 
5 7 56 
6333 
6910 
7487 
8062 
863 7 
9211 
9784 
o356 


5235 
58i3 
6391 
6968 
7544 
8119 
8694 
9268 
9841 
o4i3 


5293 
58 7 i 
6449 
7026 
7602 
8177 
8 7 5 2 
9325 
9898 
0471 


535i 
5 9 2 9 
6507 
7083 
7 65 9 
8234 
8809 
9 383 
9956 
o528 


5409 
5987 
6564 
7141 
7717 
8292 
8866 
9440 
••i3 
o585 


5466 
6o45 
6622 
7199 
7774 
8349 
8924 
9497 
••70 
0642 


5524 
6102 
6680 
7256 
7 832 
8407 
8981 
9 555 
•127 
0699 


5582 
6160 
6 7 3 7 
73i4 
7889 
8464 
9039 
9612 
•i85 
0756 


58 

58 
58 
58 
58 
57 
57 
57 

57 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 



LOGARITHMS OF NUMBEBS. 



13 



N. 





1 


2 


8 


4 


5 


6 


•7 


8 


9 


D. 


760 


880814 


0871 


0928 


0985 


1042 


1099 


Ii56 


I2l3 


1271 


i3 2 8 


5 7 


761 


1 385 


1442 


1499 


i556 


i6i3 


1670 


1727 


1784 

2354 


1841 


1898 


57 


762 


1955 

2025 


2012 


2069 
2638 


2126 


2i83 


2240 


2297 
2866 


241 1 


2468 


5 7 


763 


258i 


2695 


2752 


2809 


2923 


2980 


3o37 


5 7 


764 


3093 


3i5o 


3207 
3 77 5 


3264 


3321 


3377 


3434 


3491 
4059 
4625 


3548 


36o5 


5 7 


765 


366i 


3 7 i8 


3832 


3888 


3o45 


4002 


4u5 


4172 


57 


766 


4229 


4285 


4342 


4399 
4965 


4455 


45i2 


456 9 


4682 


4739 


57 


767 


4795 


4852 


4909 


5022 


5078 


5x35 


5192 
5 7 5 7 


5248 


53o5 


57 


768 


536i 


54i8 


5474 


553 1 


558 7 


5644 


5700 


58i3 


5870 


57 


769 


5926 


5 9 83 


6039 


6096 


6i52 


6209 


6265 


632i 


6378 


6434 


56 


770 


886491 
7054 


6547 


6604 


6660 


6716 


6 77 3 


6829 


6885 


6942 
7Do5 


6998 


56 


771 


7111 


7167 


7223 


7280 


7 2 3 S 


it 


7449 
801 1 


7661 


56 


772 


7617 
8179 


7674 
8236 


77 3o 
8292 
8853 


7786 


7842 
8404 


7898 


8067 


8i23 


56 


773 


8348 


8460 


85 7 3 


8629 


8685 


56 


774 


8741 


8797 
9 358 


8909 


8o65 
9626 


9021 


llll 


9i34 


9190 


9246 


56 


775 


9302 


94U 


9470 


9582 


9694 


975o 


9806 


56 


776 


9862 


9918 


o533 


••3o 


••86 


•141 


•107 

0756 
i3i4 


•253 


•309 
0868 


•365 


56 


777 


890421 


0477 
io35 


0589 


o645 


0700 


0812 


0924 


56 


778 


0980 
i53 7 


1091 


1 147 
1705 


1203 


1259 


1370 


1426 


1482 


56 


779 


i5 9 3 


1649 


1760 


1816 


1872 


1928 


1983 


2039 


56 


780 


892095 
265i 


2i5o 


2206 


2262 


2317 


23 7 3 


2429 


2484 


2540 


25g5 


* 56 


781 


2707 


2762 


2818 


2873 


2929 


2985 


3 040 


3096 


3i5i 


56 


782 


3207 


3262 


33i8 


33 7 3 


3429 


3484 


3 340 


3595 


365i 


3706 


56 


783 


3762 


38i 7 


38 7 3 


3928 


3984 


4039 
4093 


4094 


4160 


42o5 


4261 


55 


784 


43 16 


4371 


4427 


4482 


4538 


4648 


4704 


4759 


4814 


55 


785 


4870 


4925 


4980 


5o36 


5091 


5i46 


5201 


5257 


53i2 


5367 


55 


786 


5423 


5478 


5533 


5588 


5644 


56o 9 


5754 


5809 


5864 


5920 


55 


787 


5o 7 5 
6526 


6o3o 


6o85 


6140 


6195 


6261 


63o6 


636 1 


6416 


6471 


55 


788 


658i 


6636 


6692 


6747 


6802 


685 7 


6912 


6967 
75i 7 


7022 


55 


789 


7077 


7i32 


7187 


7242 


7297 


7352 


7407 


7462 


7572 


55 


790 


897627 


7682 


7737 


7792 


7847 


7902 


85o6 


8012 


8067 


8122 


55 


791 


8176 


823i 


8286 


8341 


8396 


845i 


856i 


86i5 


8670 


55 


792 


8725 


8780 
9328 


8835 


8890 


8944 


8999 
9 547 


9054 


9109 


9164 


9218 


55 


793 


9273 


g383 


9437 


9492 
••39 


9602 


9 656 


9711 


9766 


55 


794 


9821 


9875 


9930 


9985 


••94 


•149 


•203 


•258 


•3l2 


55 


795 


900367 


0422 


0476 


oo3i 


o586 


0640 


0695 


0749 


0804 


o85 9 


55 


796 


0913 


0968 


1022 


1077 


ii3i 


1 186 


1240 


1295 


1 349 


1404 


55 


797 


1458 


i5i3 


1 567 


1622 


1676 


1731 


1785 


1840 


1894 
2438 


1948 


54 


798 


2003 


2057 


2112 


2166 


2221 


2275 


2329 
2873 


2384 


2492 


54 


799 


2547 


2601 


2655 


2710 


2764 


2818 


2927 


2981 


3o36 


54 


800 


903090 
3633 


3 144 


3199 


3253 


3307 


336i 


34i6 


3470 


3524 


3578 


54 


801 


3687 


374i 


3795 


3849 


3904 


3 9 58 


4012 


4066 


4120 


54 


802 


4174 


4229 


4283 


433 7 


4391 


4445 


4499 


4553 


4607 


4661 


54 


803 


4716 


4770 


4824 


4878 


4932 


4986 


5o4o 


5094 
5634 


5i48 


5202 


54 


804 


5256 


53io 


5364 


5418 


5472 


5526 


558o 


5688 


5742 


54 


805 


5796 


585o 


5904 


5 9 58 


6012 


6066 


6119 


6i 7 3 


6227 


6281 


54 


806 


6335 


638 9 


6443 


6497 


655i 


6604 


6658 


6712 


6766 


6820 


54 


807 


6874 


6927 


6981 
75i 9 


7o35 


7089 


7U3 


7196 
7734 
8270 


725o 


73o4 


7358 


54 


808 


74u 


7465 


7 5 7 3 
8110 


7626 


7680 


7787 


7841 


7895 
843 1 


54 


809 


7949 


8002 


8o56 


8i63 


8217 


8324 


83 7 8 


54 


810 


908485 


853 9 


8592 


8646 


8699 


8 7 53 


8807 


8860 


8914 


8967 


54 


811 


9021 


9074 


9128 


9181 


9235 


9289 


9342 


9396 
9930 


9449 


95o3 


54 


812 


9 556 


9610 


9 663 


9716 


9770 


9823 


9877 


9984 
o5i8 


••37 


53 


813 


910091 


0144 


0197 


025l 


o3o4 


o358 


041 1 


0464 


0571 


53 


814 


0624 


0678 


0731 


0784 


o838 


0891 


0944 


0998 


io5i 


1 104 


53 


815 


n58 


1211 


1264 


i3i7 


1371 


1424 


1477 


i53o 


1584 


1637 


53 


816 


1690 


1743 


1797 


i85o 


1903 


1956 


2009 


2o63 


2116 


2169 


53 


817 


2222 


2275 


2328 


238i 


2435 


2488 


254i 


25g4 


2647 


2700 


53 


818 


2 7 53 


2806 


2859 


2913 


2966 


3019 


3072 


3i25 


3178 


323 1 


53 


819 

N. 


3284 



3337 


3390 


3443 


3496 


3549 


36o2 


3655 


3708 


3761 


53 


1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 



14 



LOGARITHMS OF NUMBERS. 



N. 





1 


♦2 


3 


4 


5 


6 


7 


8 


9 


D. 


820 


9i38i4 


386 7 


3920 


3973 


4026 


4079 

4608 


4i32 


4184 


423 7 
4766 


4290 


53 


821 


4343 


4396 


4449 


45o2 


4555 


4660 


47i3 


4819 


53 


822 


4872 


4925 


4977 


5o3o 


5o83 


5i36 


5189 


524i 


5294 


5347 


53 


823 


5400 


5453 


55o5 


5558 


56u 


5664 


5 7 i6 


5769 


5822 


5875 


53 


824 


5927 


5 9 8o 


6o33 


6o85 


6i38 


6191 


6243 


6296 


6349 

68 7 5 


6401 


53 


825 


6454 


65o7 


655 9 


6612 


6664 


6717 


6770 


6822 


6927 


53 


826 


6980 
7606 


7 o33 


7085 


7 i38 


7190 


7243 


7295 


7348 


74oo 


7453 


53 


827 


7558 


7611 


7 663 


7716 


7768 


7820 


7873 


7925 
845o 


7978 
85o2 


52 


828 


8o3o 


8o83 


8i35 


8188 


8240 


8293 


8345 


83 9 7 


52 


829 


8555 


8607 


865 9 


8712 


8764 


8816 


8869 


8921 


8 97 3 


9026 


52 


830 


919078 


9i3o 


9i83 


9235 


9287 


9340 


9 3 9 2 


9444 


9496 


9549 


52 


831 


9601 


9 653 


9706 


97 58 


9810 


9862 


9914 


9967 


••19 


••71 


52 


832 


920123 


0176 


0228 


0280 


o332 


o384 


0436 


0489 


o54i 


0593 


52 


833 


0645 


0697 


0749 


0801 


o853 


0906 


0958 


IOIO 


1062- 


1 1 14 


52 


834 


1 166 


1218 


1270 


1322 


i374 


1426 


1478 


i53o 


1582 


1 634 


52 


835 


1686 


1738 


1790 


1842 


1894 


1946 


1998 


2o5o 


2102 


2i54 


52 


836 


2206 


2258 


23lO 


2362 


2414 


2466 


25i8 


2570 


2622 


2674 


52 


837 


2725 


2777 


2829 


2881 


2 9 33 


2985 
35o3 


3o37 


3089 


3i4o 


3192 


52 


838 


3244 


3296 


3348 


3399 


345 1 


3555 


3607 


3658 


3710 


52 


839 
840 


3762 


38U 


3865 


3917 


3969 


4021 


4072 


4124 


4176 


4228 


52 


924279 


433 1 


4383 


4434 


4486 


4538 


4589 


4641 


4693 


4744 


52 


841 


479° 


4848 


4899 


4951 


5oo3 


5o54 


5io6 


5i5 7 


5209 


526i 


52 


842 


53i2 


5364 


54i5 


5467 


55.18 


5570 


562i 


5673 


5725 


5 77 6 


52 


843 


5828 


58 7 9 


5931 


5982 


6o34 


6o85 


6i3 7 


6188 


6240 


6291 


5 1 


844 


6342 


63o4 


6445 


6497 


6548 


6600 


665 1 


6702 


6754 


68o5 


5i 


845 


6857 


6908 


6969 


701 1 


7062 


7114 


7i65 


7216 


7268 


7319 


5i 


846 


7370 


7422 


7473 


7524 


7 5 7 6 


7627 


7678 
8191 


773o 


7781 


7832 


5i 


847 


7 883 


7935 


7986 


8037 


8088 


8140 


8242 


8293 


8345 


5i 


848 


8396 


8447 


8498 


8549 


8601 


8652 


8703 


8754 


88o5 


885 7 
9 368 


5i 


849 


8908 


8 9 5 9 


9010 


9061 


9112 


9163 


9215 


9266 


9317 


5i 


850 


929419 


9470 


9521 


9572 


9623 


9674 


9725 


9776 


9827 


9879 


5i 


851 


9930 


9981 


••32 


••83 


•i34 


•i85 


•236 


•287 


•338 


•38 9 
0898 


5i 


852 


93o44o 


0491 


o542 


0592 


o643 


0694 


0745 


0796 


0847 


5i 


853 


0949 


1000 


io5i 


IIQ2 


n 53 


1204 


1254 


i3o5 


i356 


1407 


5i 


854 


1458 


1 509 


]56o 


l6lO 


1661 


1712 


1763 


1814 


1 865 


1.9 1 5 


5! 


855 


1966 


2017 


2068 


2Il8 


2169 


2220 


2271 


2322 


2372 


2423 


5i 


856 


2474 


2524 


2575 


2626 


2677 


2727 


2778 


2829 
3335 


2879 


2930 


5i 


857 


2981 


3o3i 


3o82 


3i33 


3i83 


3234 


3285 


3386 


3437 


5! 


858 


3487 


3538 


358 9 


3639 


3690 


3740 


3791 


3841 


38 9 2 


3943 


5i 


859 


3993 


4044 


4094 


4145 


4195 


4246 


4296 


4347 


4397 


4448 


5i 


860 


934498 


4549 


4599 


465o 


4700 


475i 


4801 


4852 


4902 


4953 


5o 


861 


5oo3 


5o54 


5 104 


5i54 


52o5 


5255 


53o6 


5356 


5406 


5457 


5o 


862 


55o7 


5558 


56o8 


5658 


5709 


5759 


58o 9 


586o 


5910 


5960 


5o 


863 


601 1 


6061 


61 1 1 


6162 


6212 


6262 


63i3 


6363 


64i3 


6463 


5o 


864 


65 1 4 


6564 


6614 


6665 


6715 


6765 


68i5 


6865 


6916 


6966 


5o 1 


865 


7016 


7066 


7117 
7618 


7167 


7217 


7267 


73i 7 


736 7 


74i8 


7468 


5o 


866 


75i8 


7568 


7668 


7718 


7769 
8269 


7819 


7869 


7919 


7969 


5o 


867 


8019 


8069 


8119 


8169 


8219 


8320 


8370 


8420 


8470 


5o 


868 


8520 


85 7 o 


8620 


8670 


8720 


8770 


8820 


8870 


8920 


8970 


5o 


869 
870 


9020 


9070 


9120 


9170 


9220 


9270 


9320 


9369 


9419 


9469 


5o 


939619 

9400 1 8 


9569 


9619 
0118 


9660 


9719 
0218 


9769 


9819 


9869 


9918 


9968 


5o 


871 


0068 


0168 


0267 
0765 


o3n 


0367 


0417 


0467 


5o 


872 


o5i6 


o566 


0616 


0666 


0716 


o8i5 


o865 


0915 


0964 


5o 


873 


1014 


1064 


1114 


u63 


I2l3 


1263 


i3i3 


i362 


1412 


1462 


5o 


874 


1 5i 1 


i56i 


161 1 


1660 


1710 


1760 


1809 


i85 9 
2355 


1909 


iq58 


5o 


875 


2008 


2o58 


2107 


2157 


2207 


2256 


23o6 


24o5 


2455 


5o 


876 


25o4 


2554 


26o3 


2653 


2702 


2752 


2801 


285i 


2901 


2960 


5o 


877 


3ooo 


3o4q 


3099 


3i48 


3198 


3247 


3297 


3346 


3396 


3445 


49 


878 


3495 


3544 


35 9 3 


3643 


3692 


3742 


3791 


384i 


3890 


3939 


49 


879 


. 3989 


4o38 


4088 


4i37 


4186 


4236 


4285 


4335 


4384 


4433 


49 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 



LOGARITHMS OF NUMBERS. 



15 



N. 





1 


2 


3 


4 


5 


6 


. 7 


8 


9 


D. 


880 
881 
882 
883 
884 
885 
886 
887 
888 
889 

890 
891 
892 
893 
894 
895 
896 
897 
898 
899 


944483 
4976 
5469 
5961 
6452 
6943 
7434 
7924 
841 3 
8902 


4532 
5o25 
55i8 
6010 
65oi 
6992 
7483 
7973 
8462 
8g5i 


458 1 
5074 
556 7 
6059 
655i 
7041 
7532 
8022 
85n 
8999 


463 1 
5i24 
56i6 
6108 
6600 
7090 
,758i 
8070 
856o 
9048 


4680 
5i 7 3 
5665 
6i5 7 
6649 
7140 
763o 
8119 
8609 
9097 


4729 

5222 

5 7 i5 
6207 

6698 
7189 
7679 
8168 

8657 
9146 


4779 
5272 
5764 
6256 
6747 
7 238 
7728 
8217 
8706 
9195 


4828 
5321 
58i3 
63o5 
6796 
7287 

7777 
8266 
8 7 55 
9244 


4877 
5370 
5862 
6354 
6845 
7 336 
7826 
83i5 
8804 
9292 


4927 

5419 
5912 
64o3 
6894 
7 385 
7875 
8364 
8853 
9341 


49 

49 
49 
49 
49 
49 
49 
49 
49 
49 


949390 
9878 

95o365 
o85i 
i338 
1823 
23o8 
2792 
3276 
3760 


9439 
9926 
0414 
0900 
1386 
1872 
2356 
2841 
3325 
38o8 


9488 
9975 
0462 
0949 
1435 
1920 
24o5 
2889 
3373 
3856 


9 536 
••24 
o5u 
0997 
1483 
1969 
2453 
2g38 
3421 
3905 


9 585 
•• 7 3 
o56o 
1046 
i532 
2017 

2502 
2986 
3470 

3 9 53 


9634 
•121 
0608 
1095 
i58o 
2066 
255o 
3o34 
35i8 
4001 


9 683 
•170 
0657 
1 143 
1629 
2114 
2599 
3o83 
3566 
4049 


9731 
•219 
0706 
1 192 
1677 
2i63 
2647 
3i3i 
36i5 
4098 


9780 
•267 
0754 
1240 
1726 
221 1 
2696 
3i8o 
3663 
4U6 


9829 
•3i6 
o8o3 
1289 
1775 
2260 
2744 
3228 
371 1 
4194 


49 
49 
49 
49 
49 
48 
48 
48 
48 
48 


900 
901 
902 
903 
904 
905 
906 
907 
908 
909 


954243 
4725 
5207 
5688 
6168 
6649 
7128 
7607 
8086 
8564 


4291 

5255 
5 7 36 
6216 
6697 
7176 
7 655 
8i34 
8612 


4339 
4821 
53o3 
5784 
6265 
6745 
7224 
7703 
8181 
865 9 


438 7 
4869 
535i 
5832 
63i3 
6 79 3 
7272 
7 7 5i 
8229 
8707 


4435 
4918 
5399 
588o 
636 1 
6840 
7320 

7799 
8277 
8755 


4484 
4966 

5447 
5928 
6409 
6888 
7368 

7847 
8325 
88o3 


4532 
5oi4 
5495 
5 97 6 
6457 
6 9 36 
74i6 
7894 
8373 
885o 


458o 
5o62 
5543 
6024 
65o5 
6984 
7464 
1942 
8421 
8898 


4628 
5iio 
5592 
6072 
6553 
7032 

*)5l2 

7990 
8468 
8946 


4677 
5i58 
5640 
6120 
6601 
7080 
7 55 9 
8o38 
85i6 
8994 


48 

48 
48 
48 
48 
48 
48 
48 
48 
48 


910 
911 
912 
913 
914 
915 
916 
917 
918 
919 


959041 
95i8 
999 5 

96047 1 
0946 
1421 
i8 9 5 
2369 
2843 
33i6 


9089 
9 566 
••42 
o5i8 
0994 
1469 
1943 
2417 
2890 
3363 


9137 
9614 
••90 
o566 
1041 
i5i6 
1990 
2464 
2 9 3 7 
34io 


9i85 
9661 
•i38 
o6i3 
1089 
1 563 
2o38 

25ll 

2985 
3457 


9232 
9709 
•i85 
0661 
n36 
261 1 
2o85 
2559 
3o32 
35o4 


9280 

97 5 7 
•233 
0709 
1 184 
1658 

2l32 

2606 
3o 79 

3552 


9 328 

9804 
•280 
0756 

I23l 

1706 
2180 
2653 
3i26 
3599 


9375 
9852 
•328 
0804 
1279 
1753 
2227 
2701 
3i 7 4 
3646 


9423 
9900 
•376 
o85i 
i326 
1801 
2275 
2748 

3221 

36 9 3 


9471 
9947 
•423 
0899 
i3 7 4 
1848 

2322 

2 79 5 
3268 
3741 


48 
48 
48 
48 
47 
47 
47 
47 
47 
47 


920 
921 
922 
923 
924 
925 
926 
927 
928 
929 


963788 
4260 
473 1 

5202 
5672 
6l42 
66ll 
7080 
7548 
8016 


3835 
43o7 
477« 
5249 
5 7 i 9 
6189 
6658 
7127 
75 9 5 
8062 


3882 
4354 
4820 
5296 
5766 
6236 
6705 
7173 
7642 
8109 


3929 
4401 

4872 
5343 

58i3 
6283 
6752 
7226 
7688 
8i56 


3977 
4448 

5390 
586o 
6329 
6799 
7267 

v 3 i 
8203 


4024 
4495 
4966 
5437 
5 9 o 7 
6376 
6845 
73i4 
7782 
8249 


4071 

4542 
5oi3 
5484 
5954 
6423 
6892 
736i 
7829 
8296 


4118 
4590 
5o6i 
553i 
6001 
6470 
6939 
7408 
7875 
8343 


4i65 
463 7 
5io8 
55 7 8 
6048 
65i7 
6986 
7454 
7922 
8390 


4212 
4684 

5i55 
5625 
6095 
6564 
7o33 
75oi 
7969 
8436 


47 
47 
47 
47 
47 
47 
47 
47 
47 
47 


930 
931 
932 
933 
934 
935 
936 
937 
938 
939 


968483 

8 9 5o 
94i6 
9882 
970347 
0812 
1276 
1740 

2 203 
2666 


853o 
8996 
9463 
9928 
0393 
o858 

1322 

1786 
2249 
2712 


85 7 6 
9043 
9509 
9975 
0440 
0904 
1369 
i832 
2295 
2758 


8623 
9090 
9556 
••21 
0486 
0951 
Ui5 
1879 
2342 
2804 


8670 

9i36 
9602 
••68 
o533 
0997 
1461 
1925 
2388 
285i 


8716 
9i83 
9649 
•114 
0579 
1044 
i5o8 
1971 
2434 
2897 


8 7 63 
9229 
9696 
•161 
0626 
1090 
i554 

20?8 

2481 
2943 


8810 
9276 
9742 
•207 
0672 
1137 
1601 
2064 
2527 
2989 


8856 
9323 
9789 
•254 
0719 
n83 
1647 
2110 
25 7 3 
3o35 


8903 
9 36 9 
9 835 
e 3oo 
0765 
1229 
1693 
2167 
2619 
3o82 


47 
47 
47 
47 
46 
46 
46 
46 
46 
46 


N. 





1 


2 


3 


4 


5 


6 


7 


8 


9 


D. 



16 



LOGARITHMS OF NUMBERS. 



N. 





1 


2 


8 


4 


5 


6 


7 


8 


9 


D. 


940 
941 
942 
943 
944 
945 
946 
947 
948 
949 

950 
951 
952 
953 
954 
955 
956 
957 
958 
959 


973128 
35oo 
4o5i 
45i2 
4972 
5432 
5891 
635o 
6808 
7266 


3i 7 4 
3636 
4097 
4558 
5oi8 
5478 
5 9 37 
6396 
6854 
73i2 


3220 

3682 
4143 
4604 
5o64 
5524 
5 9 83 
6442 
6900 
7358 


3266 
3 7 28 
4189 
465o 
5no 
5570 
6029 
6488 
6946 
74o3 


33i3 

3774 
4235 
4696 
5i56 
56i6 
6075 
6533 
6992 
7449 


335 9 
3820 
4281 
4742 

5202 

5662 
6121 
6579 
7 o3 7 
7495 


34o5 
3866 
4327 
4788 
5248 
5707 
6167 
6625 
7083 
7541 


345i 
3913 
4374 
4834 
5294 
5 7 53 
6212 
6671 
7129 
7586 


3497 
3959 
4420 
4880 
5340 

6717 
7175 
7632 


3543 
400 5 
4466 
4926 
5386 
5845 
63o4 
6 7 63 
7220 
7678 


46 
46 
46 
46 
46 
46 
46 
46 
46 
46 


977724 
8181 

863i 
909J 
9548 
980003 
o458 
0912 
i366 
1819 


7769 
8226 
8683 
9 i38 
9594 
0049 
o5o3 
o 9 5 7 
1411 
1864 


7815 
8272 
8728 
9184 
9639 
0094 
o54o 
ioo3 
1456 
1909 


7861 
83i 7 
8774 
9230 
9685 
0140 
0594 
1048 
i5oi 
1954 


7906 
8363 
8819 
9275 
97 3o 
oi85 
0640 
1093 
1 547 
2000 


7952 
8409 
8865 
9321 
9776 

023l 

o685 
1139 
1592 
2045 


7998 
8454 
891 1 
9 366 
9821 
0276 
0730 
1 184 
1637 
2090 


8o43 
85oo 
8 9 56 
9412 
9867 

0322 
0776 
1229 

1 683 
2i35 


8089 
8546 
9002 
9457 

X 

0821 

1275 

1728 
2181 


8i35 
85 9 i 
9047 
95o3 
9958 
0412 
0867 

l320 

1773 

2226 


46 
46 
46 
46 
46 
45 
45 
45 
45 
45 


960 
961 
962 
963 
964 
965 
966 
967 
968 
969 


982271 
2723 
3i 7 5 
3626 
4077 
4527 

4977 
5426 
58 7 5 
6324 


23i6 
2769 

3220 
36 7 I 
4122 
4572 
5022 
5471 
5920 

636 9 


2362 

2814 
3265 
3716 
4167 
4617 
5067 
55i6 
5 9 65 
641 3 


2407 
285 9 
33io 
3762 
4212 
4662 
5i 1 2 
556i 
6010 
6408 


2452 
2904 
3356 
3807 
425 7 
4707 
5i5 7 
56o6 
6o55 
65o3 


2497 
2949 
3401 
3852 
43o2 
4752 

5202 

565 1 
6100 
6548 


2543 
2994 
3446 
38 97 
4347 
4797 
5247 
56 9 6 
6144 
65 9 3 


2588 
3 040 
349i 
3§42 
4J92 
4842 
5292 
574i 
6189 
6637 


2633 
3o85 
3536 

3987 
4437 
4887 
533 7 
5 7 86 
6234 
6682 


f7 8 
3i3o 
358i 
4o32 
4482 
4932 
5382 
583o 
6279 
6727 


45 
45 
45 
45 
45 
45 
45 
45 
45 
45 


970 
971 
972 
973 
974 
975 
976 
977 
978 
979 


986772 
7219 
7666 
8n3 
856o 
9006 
945o 
9805 

990339 
0783 


6817 
7264 
7711 
8i5 7 
8604 
9049 
9494 
99 3 9 
o383 
0827 


6861 

$3 

8202 
8648 
9094 
95J9 
9983 
0428 
0871 


6906 
7 353 
7800 
'8247 
86 9 3 
9i38 
9 583 
••28 
0472 
0916 


6951 
7 3 9 8 
7845 
8291 

8737 
9i83 
9628 
••72 
o5i6 
0960 


6996 
7443 
7890 
8336 
8782 
9227 
9672 
•117 
o56i 
1004 


7040 

7488 

l 9 ii 
8826 

9272 
9717 
•161 

o6o5 
1049 


7o85 
7 532 

7979 
8425 
8871 
93 1 6 
9761 
•206 
o65o 
1093 


7i3o 
7 5 77 
8024 
8470 
8916 
936i 
9806 
•25o 
0694 
1137 


7175 
7622 
8068 
85i4 
8960 
94o5 
985o 
•294 
0738 
1182 


45 
45 
45 

45 
45 
45 
44 
44 
44 
44 


980 
981 
982 
983 
984 
985 
986 
987 
988 
989 


991226 
1669 
21 1 1 
2554 
2995 
3436 
38 77 
43i7 
4757 
5196 


1270 
1713 
2i56 
2598 
3 039 
3480 
3921 
436i 
4801 
6240 


i3i5 

1758 
2200 
2642 
3o83 
3524 
3965 
44o5 
4845 
5284 


i35g 

1802 
2244 
2686 
3127 
3568 
4009 

4449 
4889 
5328 


i4o3 
1846 
2288 
2730 
3172 
36i3 
4o53 
4493 
4933 
5372 


1448 
1890 
2333 
2774 
32i6 
3657 
4097 
4537 

4977 
5416 


1492 
1935 
23 77 
2819 
3260 
3701 
4Ui 
458i 

5021 

5460 


1 536 

1979 
2421 
2863 
33o4 
3745 
4i85 
4625 
5o65 
55o4 


i58o 

2023 

2465 

3348 
3789 
4229 
4660 
5io8 
5547 


i625 
2067 
2 509 
2951 
3392 
3833 
4273 
47i3 
5i52 
5591 


44 
44 
44 
44 
44 
44 
44 
44 
44 
44 


990 
991 
992 
993 
994 
995 
996 
997 
998 
999 


995635 
6074 
65i2 

6949 
7386 
7823 
8259 
86 9 5 
gi3i 
9 565 


5679 
6117 
6555 
6993 
743o 
7867 
83o3 
8 7 3 9 
9174 
9609 


5 7 23 
6161 
6599 
7 o3 7 
7474 
7910 
8347 
8 7 8t 
9218 
9652 


l l6 l 
62o5 

6643 

7080 

7 5i 7 

7954 

8390 

8826 

9261 

9696 


58n 
6249 
6687 
7124 
7 56i 
7998 
8434 
8869 
93o5 
9739 


5854 
6293 
6 7 3i 
7168 
7605 
8041 
8477 
8913 
9 348 
9783 


5898 

633 7 * 
6774 
7212 
7648 
8o85 
8521 
8o56 
9 3 9 2 
9826 


5942 
638o 
6818 
7255 
7692 
8129 
8564 
9000 
9435 
9870 


5 9 86 
6424 
6862 
7299 
I7 36 
8172 
8608 
9043 

9479 
99 i3 


6o3o 
6468 
6906 
7343 

7779 
8216 
8652 
9087 
9522 
9957 


44 
44 
44 
44 
44 
44 
44 
44 
44 
43 


N. 





1 


2 


8 


4 


5 


6 


7 


8 


9 


D. 



TABLE II, 



CONTAINING 



NATURAL SINES AND COSINES, 



LOGAEITHMIC SINES, COSINES, TANGENTS, AND 
COTANGENTS, 



EYERY DEGREE AND MINUTE OF THE QUADRANT. 



18 



SINES AND TANGENTS. © c 



J Rad.=1. 






Logarithms. — 


Radius = 10 10 . 




] ' 


N.sine. 1 N. cos. 


L. sine. 


D. 1" 


L. cos. D.l" 


L. tang. 1 D. 1" 


L. cot. 







00000 


Unit. 


i 0-000000 




10-000000 




o- 000000 


Infinite. 


60 


1 


00029 


Unit. 


[6-463726 


5017-17 


000000 


•00 


6-463726 5017.(17 


13-536274 59 


2 


00038 


Unit. 


764756 


2934-85 


000000 j 


•00 


764706 2934 


•83 


235244 58 


8 


! 00087 


Unit. 


940847 


2o82-3i 


000000 


•00 


940847 2082 


-3i 


009103 57 


4 


: 00116 


Unit. 


7-065786 


i6i5 • 17 
i3i 9 -68 
1 1 1 6-75 


000000 


•00 


7 •060786 i6i5 


■H 


12.934214 56 


5 


ooi45 


Unit. 


162696 


000000 ! 


•00 


162696 i3ig 


•69 


837304 55 


6 


00175 


Unit. 


241877 


9-999999! 


•01 


241878 ni5 


■ 78 


768122 154 


7 


00204 


Unit. 


308824 


966-53 


999999 


•01 


3o8825 


996 


•53 


691 1 7 5 53 
633i83'52 


8 


00233 


Unit. 


1 366816 


852-54 


999999 


•01 


3668i 7 


852 


04 


9 


00262 


Unit. 


417968 


762-63 


999999 

999998 1 

9-999998 


•01 


417970 


762 


63 


582o3o 51 


10 


00291 


Unit. 


I 463725 


689-88 
629-81 


•01 

-01 


463727 


689 


88 


536273 j 50 


-n" 


00320 


99999 


7-5o5u8 


7'5o5i2o 


629 


81 


12-494880 49 


12 


00349 
00378 


99999 


542906 


579-36 


999997 


•01 


542909 


O79 


33 


467091 48 


13 


99999 


577668 


536-41 


999997 : 


•01 


577672 


536 


42 


422328 147 


14 


00407 


99999 


609853 


499-38 


999996 


•01 


609857 


499 


39 


390143 46 


15 


oo436 


99999 


63 9 8i6 


467-14 


999996 


•01 


639820 


467 


i5 


36oi8o 45 


16 


oo465 


99999 


667845 


438-8i 


999995 


•01 


667849 


438 


82 


332i5i 144 


17 


00495 


99999 


694173 


4i3-72 


999995 


•01 


694179 


4i3 


73 


3o582i |43 


18 


OOD24 


99999 


718997 


391 -35 


999994 


•01 


719003 


391 


36 


280997 \ 42 


18 


oo553 


99998 


742477 


371-27 


999993 


•01 


742484 


3 7 i 


28 


207016 41 


20 
21 


oo582 


99998 


764754 


353-i5 


999993 


•01 


764761 


35i 


36 


235239 !40 


0061 1 


99998 


7-780943 


336-72 


9-999992 


•01 


7-785951 


336 


73 


12-214049 89 


22 


00640 


99998 


806146 


321-75 


999991 


•01 


806 1 55 


321 


76 


193845:38 


23 


00669 


99998 


82545i 


3o8-o5 


999990 
999989 
999988 


•01 


826460 


3o8 


06 


174540 37 


24 


00698 


99998 


843934 


295-47 


•02 


843944 


2 9 5 


49 


i56o56j36 


25 


00727 


99997 


861662 


283-88 


•02 


861674 


283 


90 


138326 35 


26 


00706 


99997 


878690 


273.17 


999988 


•02 


878708 


2 7 3 


18 


1 21 292 34 


27 


00780 


99997 


8 9 5o85 


263-23 


999987 


•02 


895099 


263 


25 


1 0490 1 33 


28 


00814 


99997 


910879 


253.99 
245-38 


999986 


•02 


910894 


254 


01 


089106 82 


29 


00844 


99996 


9261 19 


999985 


•02 


926134 


245 


40 


073866 31 


30 


00873 


9999 6 


940842 


23 7 -33 


999983 


•02 


94o858 
7.955100 


23 7 

229 


35 
81 


069142 80 
12-044900 29 


~3T 


00902 


99996 


7-955082 


229-80 


9-999982 


•02 


32 


00931 


99996 


968870 


222.73 


999981 


•02 


968889 


222 


75 


o3i 111 


28 


33 


00960 


99995 


982233 


216-08 


999980 


•02 


982253 


2l6 


10 


017747 


27 


34 


00989 


99995 


995198 


209-81 


999979 


•02 


995219 


209 


83 


004781 


26 


35 


01018 


o)igo5 


8-007787 


203.90 


999977 


•02 


8-007809 


203 


92 


11-992191 25 


36 


01047 1-^9995 


020021 


198-31 


999976 


•02 


020045 


198 


33 


979950 j 24 


37 


01076 


99994 


031919 


193-02 


999975 


•02 


o3 1 945 


193 

188 


o5 


968066 ! 23 


38 


ono5 


99994 


o435oi 


188-01 


999973 


•02 


043527 


o3 


956473!22 ' 


39 


ou34 


99994 


054781 


183-25 


999972 


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178-72 


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8-076500 


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1 16926 


159-08 


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768079 







N. cos. N. sine. 


L. cos. 


D. 1" 


L. sine. 


L. cot. 


D.l" 


L. tang. 


' 


§9° 



SINES AND TANGENTS.- — 1°. 



19 



Rad. = L 




Logarithms. — 


Radius = 


1010. 







N.sineJ N. cos. 


L. sine. 


D.l" 


L. cos. j 


D.1" 


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D.l" 


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9-999848 


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60 -04 


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60-12 


11-456916 


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L. cos. 


D.l" 


L. sine. 




L. cot. 


D.l" 


L. tang. I ' 


§8° 



Ji 



20 



SINES AND TANGENTS. — 2 ( 



Kad. = 1. 




Logarithms. — Radius =101°. 




t 



N. sine. 


N. cos. 


L. sine. 


D.l" 


L. cos. 


D.1" 


tt. taflg. 


D.l" 


L. cot. 




03490 


99939 


8-542819 


60-04 


9-999735 


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8- 543o84 


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719396 40-17 


280604 


I 




N. cos. ! N. sine. 


L. cos. 


D.l" 


L. sine. 




L. cot. D. V 


L. tang. 


1 jj 


§7° 



SINES AND TANGENTS. 3°. 



21 



Rad. = 1. 


Logarithm 


s.— Radius =10!0. 


~^i 


N.sineJ N. cos. 


L. sine. 


D. 1" 


L. cos. 


D.1" 

•11 


L. tang. 


D.l" 


L. cot. I 


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•i5 j 844644 30-19 


155356 | 


,N. cos. 


N. sine. 


L. cos. j D. 1" 


L. sine. 


L. cot. j D. 1" 


L. tang. | 


§6° 



22 



SINES AND TANGENTS.— 4°. 



Rad. = 1. 




Logarithms. — Radius = 10 10 . 







N. sine. 
06976 


N. cos. 


L. sine. 


D. 1" 


L. cos. 


D.l" 


L. tang. 


Dl." 


L. cot. j 


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8-843585 


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9-998941 


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67 


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9985o6 


• 18 


919568 


25 


47 


o8o432 


15 


46 


o83io 


99654 


919591 


25 


20 


998495 


•18 


921096 


25 


38 


078904 


14 


47 


08339 


99652 


921103 


25 


12 


998485 


• 18 


922619 


25 


3o 


o 77 38i 


13 


48 


o8368 


99649 


922610 


25 


o3 


998474 


-18 


924136 


25 


21 


075864 


12 


49 


08397 


99647 


924112 


24 


94 


998464 


.18 


925649 


25 


12 


074351 


11 


50 
51 


08426 


99644 


j 926609 


24 


86 


998453 


.18 


927156 
8^928658 


25 


o3 


072844 
11 -071342 


10 
9 


o8455 


99642 


18-927100 


24 


77 


9-998442 


•18 


24 


95 


52 


08484 99639 


j 928587 


24 


69 


998431 


.18 


93o 1 55 


24 


86 


069845 


8 


53 


o85i3 i 99637 


930068 


24 


60 


998421 


.18 


931647 


24 


78 


068353 


7 


54 


08542 ! 99635 


1 93 1 544 


24 


52 


998410 


• 18 


933i34 


24 


70 


066866 


6 


55 


08571 ; 99632 


j 933o 1 5 


24 


-43 


998399 
9 9 8388 


.18 


934616 


24 


61 


o65384 


5 


56 


08600 99630 


93448i 


24 


• 35 


• 18 


; 936093 


24 


53 


063907 


4 


57 


08629 99627 


935942 


24 


•27 


998377 


• 18 


9 37565 


24 


45 


062435 


3 


58 


o8658 j 99625 


937398 


24 


• 19 


998366 


• 18 


939032 


24 


37 


060968 
o595o6 


2 


59 1 08687 i 99622 


9 3885o 


24 


•11 


99 8355 


• 18 


1 940494 


24 


3o 


1 


60 08716 99619 


940296 


24 -o3 


998344 


.18 


941 9 52 


24-21 


o58o48 





|JN. cos. IN. sine. 


1 L. cos. 


D. 1" 


L. sine. 




1 L. cot. 


D.l" 


L. tang. 


' 


§5° 



SINES AND TANGENTS. 5°. 



23 



Rad. = 1. 






Logarithms. — 


Kaditts — 


10 10 . 






N.sine. N. cos. 


L. sine. 


D. 1" 


L. cos. \~D.1" 


L. tang. 


D. 1" 


L. cot. 




08716 99619 


8-940296 
94H38 


24 


•o3 


9-998344! -19 


8-941962 


24-21 


u-o58o48 


60 


1 1 08743 99617 


23 


94 


998333 i 


-19 


943404 


24 


i3 


066696 


59 


2 ; 08774; 99614 


943174 


23 


87 


998322 


•19 


944862 


24 


o5 


o55i48 


58 


3 1 o88o3' 99612 


944606 


23 


79 


9983 1 1 


•19 


946296 


23 


97 


063705 


57 


4 


08801 99609 


946034 


23 


7' 


998300 


.19 


947734 


23 


90 


o52266 


56 


5 


08860 ! 99607 


947456 


23 


63 


998289 


•19 


949168 


23 


82 


o5o832 


55 


6 


08889 
08918 


99604 


948874 


23 


55 


998277 


.19 


950697 


23 


74 


049403 


54 


7 


99602 


960287 


23 


48 


998266 


•19 


962021 


23 


66 


047979 
046559 


53 


8 


08947 


99599 


951696 


23 


40 


998255 


.19 


953441 


23 


60 


52 


9 


08976 99396 


963100 


23 


32 


998243 


•19 


954856 


23 


5i 


046144 


51 


10 

IT 


09005 1 99394 
09034 ! 99691 


954499 
8 •955894" 


23 


25 


998232 


•19 
•19 


956267 


23 


44 


043733 


50 


23 


17 


9-998220 


8-967674 


23 


37 


11-042326 


49~ 


12 


09063 ' 99388 


937284 


23 


10 


998209 


•19 


959075 


23 


29 


040926 
039627 


48 


18 


09092 99386 


968670 


23 


02 


998197 


•19 


960473 


23 


23 


47 


14 


09121 99 583 


960062 


2 2 


95 


998186 


•19 


961866 


23 


14 


o38i34 


46 


15 


09150 j 99380 


961429 


22 


88 


998174 


•19 


963255 


23 


07 


o36745 


45 


16 


09179 


99 5 7 8 


962801 


22 


80 


998163 


.19 


964639 


23 


00 


o3536i 


44 


17 


09208 


99575 


964170 


22 


73 


998151 


.19 


966019 


22 


9 3 


033981 


43 


18 


09237 


99372 


o65534 


22 


66 


998139 


• 20 


967394 


22 


86 


032606 


42 


19 


09266 


99570 


966893 


22 


5 9 


998128 


-20 


968766 


22 


79 


o3i234 


41 


20 

21 


09293 


99567 


968249 
8 • 969600 


22 


32 


9981 16 


•20 
•20 


970l33 


22 


7i 


029867 


40 


09324 99564 


22 


44 


9-998104 


8-971496 


22 


65 


11 -028604 


39 


22 


09353 i 99362 


970947 


22 


38 


998092 


•20 


972855 


22 


5 7 


027145 


38 


23 


09382 | 99359 


972289 


22 


3i 


998080 


•20 


974209 


22 


5i 


025791 


37 


24 


o94i 1 99356 


973628 


22 


24 


998068 


•20 


976660 


22 


44 


022440 


36 


25 


09440 9 9 553 


974962 


22 


17 


998056 


•20 


976906 


22 


37 


023094 


35 


26 


09469 


9955i 


976293 


22 


10 


998044 


•20 


978248 


22 


3o 


021762 


34 


27 


09498 


99548 


977619 


22 


o3 


998032 


•20 


979686 


22 


23 


020414 


33 


28 


09527 


99545 


978941 


21 


97 


998020 


•20 


980921 


22 


17 


019079 


32 


29 


09556 


99542 


980259 


21 


S 


998008 


•20 


982251 


22 


10 


017749 


31 


30 

~3T 


09583 
09614 


99540 
99537 


981573 


21 


997996 


•20 


983577 


22 


04 


016423 


30 


8-982883 


21 


77 


9-997984 


•20 


8-984899 


21 


97 


ii-oi5ioi 


29 


32 


09642 


99534 


984189 


21 


70 


997972 


•20 


986217 


21 


8! 


013783 


28 


33 


09671 


9953i 


985491 


21 


63 


997959 


•20 


987532 


21 


012468 


27 


34 


09700 


99628 


986789 
988083 


21 


57 


997947 


■20 


988842 


21 


78 


oiu58 


26 


35 


09729 99526 


21 


5o 


997 9 35 


•21 


990149 


21 


7 i 


009861 
008649 


25 


36 


09758 99523 


989374 


21 


44 


997922 


•21 


99i45l 


21 


65 


24 


37 


09787 | 99520 


990660 


21 


38 


997910 


•21 


992750 


21 


58 


007260 


23 


38 


09816 99517 


991943 


21 


3i 


997897 


•21 


994o45 


21 


52 


006955 


22 


39 


09845 99514 


993222 


21 


25 


997886 


•21 


995337 


21 


46 


004663 


21 


40 
41 


09874 9951 1 


994497 
8-995768 


21 


!9 


997872 


•21 


996624 


21 
21 


40 


003376 


20 


09903 


995o8 


21 


12 


9-997860 


•21 


8-997908 


34 


11 -002092 


19 


42 


09932 


99506 


997036 


21 


06 


997847 


•21 


999188 


21 


27 


000812 


18 


43 


09961 


995o3 


998299 


21 


00 


997835 


•21 


9-000466 


21 


21 


10-999535 


17 


44 


09990 


99600 


999560 


20 


94 


997822 


•21 


001738 


21 


i5 


998262 


16 


45 


1 00 1 9 


99497 


9-000816 


20 


o 7 


997809 


•21 


003007 


21 


3 


996993 


15 


4 S 


10048 


99494 


002069 


20 


82 


997797 


•21 


004272 


21 


995728 


14 


47 


10077 


99491 


oo33i8 


20 


76 


997784 


•21 


oo5534 


20 


97 


994466 


13 


48 


1 01 06 99488 


004563 


20 


70 


99777' 


•21 


006792 


20 


U 


993208 


12 


49 


ioi35 99485 


oo58o5 


20 


64 


997758 


•21 


008047 


20 


991953 


11 


50 
~5T 


10164 99482 


007044 


20 


58 


997743 


•21 


009298 


20 


80 


990702 


10 


10192 99479 


9-008278 


20 


52 


9-997732 


•21 


9-oio546 


20 


74 


10-989464 


9 


52 


10221 


99476 


009610 


20 


46 


997719 


•21 


01 1790 


20 


68 


988210 


8 


53 


10250 


99473 


010737 


20 


4o 


997706 


•21 


oi3o3i 


20 


62 


986969 


7 


54 


10279 


99470 


01 1962 


20 


34 


997693 


•22 


014268 


20 


56 


985732 


6 


55 


io3o8 


99467 


oi3i82 


20 


29 


997680 


•22 


oi55o2 


20 


5i 


984498 


5 


56 


io337 


99464 


014400 


20 


23 


997667 


•22 


016732 


20 


45 


983268 


4 


57 


io366 


9946i 


oi56i3 


20 


17 


997654 


•22 


017959 


20 


40 


982041 


3 


5S 


io395 99458 


016824 


20 


12 


997 6 4i 


•22 


019183 


20 


33 


980817 


2 


59 


10424 


99455 


oi8o3i 


20 


06 


997628 


•22 


020403 


20 


28 


979697 
978380 


1 


60 


10453 


99452 


019235 


20-00 


997614 


•22 


021620 


20-23 







j N. cos. |N. sine. 


L. cos. 


D. 1" 


L. sine. 




L. cot. 


D. 1" 


L. tang. 


' 


§4° 



24 



BINES AND TANGENTS. 6°. 



Rad. = 1. Logarithms. — Radius = 10'°. 


t 


1 
2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 

21" 

22 
23 
24 
25 
26 
27 
28 
29 
30 
"31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
4S 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


N. sine. N. cos. L. sine. 


D.l" 


L. cos. 


D.l" 


L. tang. 


D. 1" 


L. cot. 




10453 99452 
10482 99449 
io5ii 99446 
io54o 99443 
10569 i 9944o 
10397 99437 
10626 : 99434 
io655 99431 
10684 1 99428 
10713 99424 
10742 99421 


9-019235 
020435 

021632 

022825 
024016 

025203 

026386 
027067 
028744 
029918 
031089 


20 
19 
19 
19 
x 9 
19 
l 9 

*9 

l 9 
J 9 
19 


00 
9 5 

59 
84 
78 
73 
67 
62 

5i 

47 


9-997614 
997601 
997588 
997574 
997061 
997547 
997 5 34 
997520 
997307 
997493 
997480 


•22 
•22 
•22 
•22 
• 22 
•22 

-23 
•23 
•23 
•23 
•23 


9-021620 
022834 
024044 

02525l 

026455 
027655 

028802 

o3oo46 
o3i237 
o32425 
033609 


20 
20 
20 
20 
20 
10 
19 
'9 
19 
19 
19 


23 

17 
11 

06 
00 
95 
90 

85 
79 
74 
69 


10-978380 
977166 
975 9 56 

974749 
973545 
972345 
971U8 
969954 
968763 
967575 
966391 


60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42 
41 
40 


10771 
10800 
10829 
io858 
10887 
10916 
10945 
10973 
1 1002 
no3i 


99418 
9941 5 
99412 
99409 
99406 
99402 
99399 
99396 
99393 
99390 


9^o32257 
o3342i 
o34582 
035741 
o368g6 
o38o48 
039197 
040342 
04i485 
042625 


l 9 
19 
19 
19 
19 
19 
19 

18 


41 
36 
3o 

25 

20 
i5 
10 
o5 

99 
94 


9-997466 
997452 
997439 
997425 
99741 1 

997397 
997383 
997369 
997355 
997341 


•23 
•23 
•23 
•23 
•23 
•23 
•23 

•33 

•23 
•23 


9-034791 
035969 
037144 
o383i6 
039485 
0406 5 1 
041813 
042973 
0441 3o 
040284 


l 9 
19 
*9 
19 
19 
19 
*9 
l 9 
l 9 
J 9 


64 
58 
53 
48 
43 
38 
33 
28 

23 

18 


10-965209 
964o3i 
962856 
961684 
9605 1 5 
959349 
9 58i8 7 
957027 
955870 
954716 


1 1060 
1 1 089 
11118 

n 147 
11 176 

II205 

11234 
1 1 263 
11291 

Il320 

1 1 349 

n3 7 8 
1 1407 
1 U36 
1 1465 
1 1 494 
u523 
n552 
11 586 
1 1 609 

7i638 

1 1 667 
1 1696 
11725 
II754 
n 7 83 
11812 
1 1 840 
1 1 869 
1 1 898 

1 1956 
1 1985 
12014 
12043 
1 207 1 
1 2 100 
1 21 29 
i2i58 
12187 


99 386 
99383 
99380 
99377 
99374 
99370 
99367 
99364 
99360 
99337 
99354 
9935i 
99347 
99344 
99341 
99337 
99334 
9933i 
99327 
99324 
99320 
99317 
99314 
99310 
99307 
993 o3 
99300 
99297 
99293 
99290 


9-043762 
044895 
046026 
047154 
048279 
049400 
o5o5i9 
o5i635 
052749 
o53859 

9-054966 
056071 
057172 
058271 
059367 
060460 
o6i55i 
062639 
063724 
064806 


18 
18 
18 
18 

18 
18 
18 
18 
18 
18 


8 9 
84 
79 
7 D 
70 
65 
60 
55 
5o 
45 


9-997327 
9973 1 3 

997299 
997285 
997271 
997257 
997242 
997228 
997214 
99/199 


•24 
•24 
•24 
•24 

• 24 

• 24 

• 24 
•24 
•24 

•24 


9 • 046434 
047582 
048727 
049869 
o5ioo8 
o52i44 
053277 
054407 
055535 
o56659 


l 9 
19 

\l 

18 
18 
18 
18 
18 
18 


i3 

08 
o3 

98 
93 

?9 
84 

79 
74 
70 


10-953566 
952418 
951273 
95oi3i 
948992 
947856 
946723 
945593 
944465 
943341 


39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 


18 
18 
18 
18 
18 
18 
18 
18 
18 


41 
36 
3,i 

27 
22 

i3 
08 
04 
99 


9-997185 
997170 
997 1 56 
997141 
997127 
997112 
997098 
997083 
997068 
997053 


•24 
•24 
•24 
•24 
•24 
•24 
•24 

•25 
•25 
•25 


9-057781 
058900 
060016 
061 i3o 
062240 
o63348 
o64453 
965556 
o66655 
067752 


18 
18 
18 
18 
18 
18 
18 
18 
18 
18 


65 

69 
55 
5i 
46 
42 
37 
33 
28 
24 


10-942219 
941 100 
939984 
938870 
937760 
936652 
935547 
934444 
933345 
932248 


9 .o65885 
066962 
o68o36 
069107 
070176 
071242 
072306 
073366 
074424 
075480 


17 

17 

17 
17 
17 
17 


94 
90 
86 
81 

77 
72 
68 
63 

n 


9.997039 
997024 
997009 
996994 
996979 
996964 
996949 
996934 
996919 
996904 


•25 
•25 
•25 
•25 
•25 
•25 
•25 
•25 
•25 
•25 
•25 
•25 
•25 
•25 
•25 

.26 
.26 
.26 
.26 
.26 


9-068846 
069938 
071027 
072113 
073197 
074278 
075356 
076432 
077505 
o 7 85 7 6 


18 
18 
18 
18 

18 
17 
17 

17 


10 
06 
02 

97 

t 

84 
80 


1 • 93 1 1 54 
930062 
928973 
927887 
926803 
925722 
924644 
923568 
922495 
921424 


19 
18 
17 
16 
15 
14 
13 
12 
11 
10 


99286 
99283 
99279 
99276 
99272 
99269 
99265 
99262 
99258 
99255 


9-076533 
077583 
078631 
079676 
080719 
081759 
082797 
o83832 
084864 
085894 


17 

17 
17 

17 

17 


5o 
46 
42 
38 
33 
29 

2D 
21 

17 

i3 


9-996889 
906874 
99 6858 
996843 
996828 
996812 

996797 
996782 
996766 
996751 


9-079644 
080710 
081773 
082833 
083891 
084947 
086000 
087050 
088098 
089144 


17 

17 
17 
17 
»7 
17 
17 


76 
72 
67 
63 
5 9 
55 
5i 

47 
43 
38 


io-92o356 
919290 
"918227 
917167 
916109 
9i5o53 
914000 
912950 
91 1902 
9io856 


9 
8 
7 
6 
5 
4 
3 
2 
1 





N. cos. 


N. sine. 


L. cos. 


D.l" 


L. sine. 




L. cot. 


D.l" 


L. tang. 


1 


83° 



SINES AND TAJSGtENTS. 7' 



25 



J ■ ■ " 

1 .Rad. = 1. 


Logarithms. — Eadius = 10'°. 


1 ; 


N. siBe.: N". cos. 


L. sine. 


D. 1" | L. cos. 


D.l" 


L. tang. 


D. 1" 


L. coc. 




j ° 

1 2 

4 

1 6 

I 7 

1 8 
1 9 

10 

fir 

12 
13 

14 
15 
16 

17 
18 
19 



21 
22 
23 
24 
25 
26 
27 
28 
29 
_80_ 
31 
32 
33 
34 
35 
36 
37 
3S 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
j 55 
1 56 

h 7 

5S 
59 
60 


12187 ' 99235 
12216 ' 99251 
12245 , 99248 
12274:99244 

I23o.2 ; 99240 

i233i ; 99237 
i236o ! 99233 
12389 1 9923o 
12418 99226 
12447 j 99222 
12476 99219 


9-083894 
086922 
087947 
088970 
089990 
091008 
092024 
093037 
094047 
093036 
096062 


:? 

*7 
*7 
16 
16 
16 
16 
16 
16 
16 


•i3 

• 09 
•04 
00 

96 
.92 
-88 
•84 
80 
76 
7 3 


9-996751 

996735 
996720 
996704 
996688 
996673 
996657 
996641 
996625 
996610 

996594 


•26 
•26 
-26 
•26 
•lb 
•26 
•26 
-26 
•26 
•26 
•26 


9-089144 
090187 
091228 
092266 
O933o2 
094336 
095367 
096395 
097422 
098446 
099468 


17-38 
17-34 
i7-3o 
17-27 
17-22 
17.19 
17- 1.5 
17. 11 
17-07 
17-03 
16-99 


10-910856 
909813 
908772 
907734 
906698 
903664 
904633 
903 6o5 
902578 
90i554 
900582 

10-899518 
898496 
807481 
896468 
895408 
894450 
893444 
892441 
891440 
890441 

10 • 889444 
888449 
887457 
886467 
885479 
884493 
883509 
882528 
88i548 
880571 


60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 
49 
48 
47 
46 
45 
44 
43 
42' 
41 
40 

39 
38 
37 
36 
35 
34 
33 
32 
31 
80 
W 
28 
27 
26 
25 
24 
23! 
22 
21 
20 


i25o4 
12533 

12562 
I25 9 I 

12620 

12649 
12678 

12706 

1.2735 
12764 


99 2i5 
99211 
99208 
99204 
99200 
99197 
99193 

99* 8 9 
99186 
99182 


9-097065 
098066 
099063 
100062 
ioio56 
102048 
108037 
104025 
io5oio 
105992 


16 
16 
16 
16 
16 
lb 
16 
16 
16 
16 


68 
65 
6i 

u 
s 

41 

38 
34 


9-996578 
996562 
996546 
996530 
996514 
996498 
996482 
996465 
996449 

996433 


•27 
■27 
•27 
•27 

•27 

•27 

'27 
•27 
-27 

•27 
•27 
•27 
•27 
•27 

• 27 

•27 
•27 
.28 
.28 

• 28 
.28 
.28 

• 28 

• 28 
.28 
.28 
.28 

• 28 
.28 
•28 


9-100487 
101304 
102519 
io3532 
104542 
io555o 
io6556 
107559 
io856o 
109559 

9- no556 
iii55i 
1 1 2543 
n3533 
ii452i 
n55o7 
116491 
117472 
u8452 
119429 


i6-o5 
16-91 
16-87 
16-84 
16-80 
16-76 
16-72 
16-69 
16. 65 
16. 61 
~i6^58~ 
16.54 
16. 5o 
16.46 
i6.43 
16.39 
16-36 
16-82 
16-29 

l6- 25 


12793 

1282.2 
1 285 1 
1288.0 
12908 
12987 
12966 
12995 
1 3 024 
i3o53 


99178 
99175 
99*7* 
99*67 
99*63 
99160 
99 1 56 
99i52 
99U8 
99*44 


9-106973 
107951 
108927 
109901 
110873 
111842 
1 1 2809 
118774 
1 14737 
n56 9 8 


16 
16 
16 
16 
16 
16 
16 
16 
16 
i5 


3o 
27 

23 

16 
12 
08 
o5 
01 
97 


9-996417 

996400 

996384 

99 6368 
99635i 
996335 
996318 
996302 
996285 
996269 


i3o8i 
i3no 
i3i39 
18168 
18197 

13226 

i3254 
i3283 
i33i2 
i334i 
13870 
13399 
13427 
13456 
i3485 
i35i4 
13543 
i35 7 2 
i36oo 
18629 

13658" 

i368 7 
18716 
18744 
i3 77 3 
i38o2 
1 383 1 
i386o 
i388 9 
18917 


99*41 

99137 
991 33 
99129 
99125 
99122 
99118 
99**4 
991 10 
.99 1 06 


9-M6656 
117613 
ii856 7 
• 119519 
120469 
121417 

122362 

i233o6 
124248 

125187 


i5 

1 5 
i5 
i5 
i5 
i5 
i5 
i5 
i5 
i5 


94 
90 

87 
83 
80 
76 

73 
69 
66 
62 


9-996252 
996235 
996219 
996202 
996185 
996168 
996151 
996184 
996117 
996100 


9-120404 
121377 
122348 
123317 
124284 
125249 
126211 
127172 
i28i3o 
129087 


l6-22 
l6-l8 

16. i5 

16. II 

16-07 
16.04 
16-01 

15.97 
15.94 

15.91 


10-879596 
878623 
877652 
876683 
875716 
874751 
873789 
872828 
871870 
870913 


99102 
99098 
99094 
99091 
99087 
•99088 
99079 
99075 
99071 
99067 
99063 
99059 
99o55 
99o5i 
99047 
99043 
99039 
99035 
9903 1 
99027 


9-126125 
127060 
127993 
128925 
129854 
130781 
181706 
i3263o 
i3355i 
134470 

9^735387 
i363o3 
137216 
i,38*28 
189037 
139944 
i4o85o 
•i4i7 5 4 
142655 
143555 


i5 
i5< 
*5- 
i5 
i5 
i5 
i5- 
i5 
i5 
i5 


5 9 
56 

52 

% 

42 
3 9 
35 

32 

29 


9-996088 
996066 
996049 
996032 
996015 
995998 
995980 
995963 
995946 
995928 


•29 
•29 
.29 

•29 

• 29 

.29 

•29 
.29 

• 29 
.29 


6-i3oo4i 
130994 
131944 
i328 9 3 
133839 
184784 
1357.26 
136667 
1876.05 
138542 


i5.8 7 
15-84 
i5-8i 
15.77 
15-74 
i5- 7 i 
15-67 
15-64 
i5-6i 
15-58 


10-869959 
869006 
868o56 
867107 
866 1 61 
8652i6 
864274 
863333 
862395 
86i458 


19 
•18 
17 
16 
15 
14 
18 
12 
11 
10 


i5 

i5 
i5 
i5 
i5 
iS 
i5 
i5 
i5- 
U 


25 

22 
19 
16 
12 
09 
06 
o3 

OO 

96 


9-995911 

993894 
995876 
995859 
995841 
995823 
995806 
995788 
995771 
995753 


•29 
•29 
•29 
.29 

•29 
.29 

•29 

.29 

•29 
.29 


9-189476 
140409 
i4i34o 
142269 
143196 
1 44i 2 1 
i45o44 
145966 
146885 
U78o3 


i5.55 
i5-5i 
i5-48 
15-45 
i5-42 
l5-3 9 
i5-35 
15-32 
i5-29 
i5-26 


io-86o524 
859591 
85866o 
85 77 3i 
8568o4 
855879 
854956 
854o34 
853n5 
852197 


9 

8 : 

7' 

6 

51 

4 

3 

2 

1 • 




M". .cos. : K sine. 

1 


L. cos. 


ixi" 


L. sine. 




L. cot. 


D.l" 


L. tang. 


' 


82° 



26 



SINES AND TANGENTS. 8°- 



Rad.=J. 


j 
Logarithms. — Eadius= I0 1(V . 




1 

2 
8 

4 
5 
6 

7 

8 

9 

10 

11 
12 
13 
14 
15 
16 
17 
.18 
19 
20 


N. sine. 


N. cos. 


L. sine-. 


D.1" 


L. cos. 


D.l" 


L. taog. 


D 1/' 


L. cot. 




13917 
13946 
i3 9 75 
14004 
14033 
1 406 1 
14090 

14119 
1 41 48 

14177 
U2o5 


99027 

99023 

99019 

99016 

990 u 

99006 

99002 

98998 

98994 

98990 

98986 

98982" 

98978 

98973 

98969 

98965 

98961 

98957 

98953 

98948 

98944 


9- 143555 
144453 
145349 
146243 
I47I36 
148026 
148915 
149802 
i5o686 
i5i569 
10245 1 

9.i5333o 
1.54208 
i55o83 
155957 
i5683o 
157700 
i5856a 
i59435 
i6o3oi 
161 164 


14 
14 

14 
14 
14 
U 

u 
14 
14 
14 
14 


96 
93 
90 

2 7 
84 

■81 

75 
72 
69 
66 


9-995753 
995735 
995717 
995699 
995681 
995664 
99.5646 
995628 
9956 1 
995591 
995573 


•3o 

• 3o 

• St* 

• 3o 
•3o 
•3o 
-3o 
•3o 
-3o 
•3o 
-3o 

~3~6~ 
•3o 
-3o 
•3i 

-3i 
•3i 
•3i 

• 3i 

• 3i 

• 3i 
•3i 

• 3i 

• 3i 

• 3i 

• 3i 

• 3i 

•32 
•32 
•32 


9,-147803 
1487 18 
149632 
1 5o544 
i5i454 
152363 
153269 
1 54 1 74 
155077 
165978 
166877 

9-157776 
158671 
1 59565 
160457 
1 61 347 
162 236 
1 63i 23 
164008 
164.892 
160774 

9-166654 
167532 
168409 
169284 
1,70157 
171029 
171899 
172767 
173634 
174499 

9- 175362 
176224 
177084 
577942 
178799 
179655 
180008 
i8i36o 

I 8 2-2:1 I 

1 88009 


16-26 
16 -23 
l5-20 

.5,17 

10-14 
i5-u 
r6-o8 
16 -o5 

l6- 02 

14.99 
14-96 


io* 852197 
85 1 282. 
85o368 
849456 
848546 
847637 
846731 
§45826 
844923 
844022 
843i23 

10-842225 
841329 
840435 
839543 
838653 
83.7764 
836877 
835992 
835 108 
834226 

BO- 833346 

832468 
831D91 

830716 
839843 
82897 i 
828101 
827233 
826366 
8255oi 


60 

59 

58 

57 

56 J 

55 

54 

53 

52 ; : 

51 
50 j 

49 : 

48 
47 
46 j 

45 ; 

44 | 
43 | 
42 
41 1 

39 ■ 

38 

37 
86 [ 

35 
34 
33 
32 
31 i 
30 ; 

29 [ 
28 
27 

26 J 
25 
24 j 
23 : 
22 
21 I 
20 
19 
18 ;. 

!• 

13 1 

12 : 

11 1 

10 ; 
9 i 

? ; l 
v\ 

M 

2 ' 
1 


14234 
14263 
14292 
14320 
14349 
14378 
14407 
U436 
14464 
14493 


14 

14 
14 
14 
14 

14 
14 
14 
14 
14 


63 
60 
5 7 
54 
5i 
48 
45 
42 
39 
36 


9.995555 
990537 
995019 
995501 
995482 
990464 
995446 
993427 
995409 
995390 


14-9.3 
J4-9P 
14-87 
H'U 
14-81 

U-79 
14-76 
14-73 
14-70 
14-67 


21 
22 
23 
24 
25 
26 
27 
28 
29 
30 


14^22 

U55i 

U58o 
14608 
i463 7 
14666 
14695 
14723 
U752 
14781 


98940 1 9.. i62025 
98936 162885 
98931 163743 
98927 | 164600 
98923 I 165454 
98919 | 166807 
98914 1 167159 
98910 | 168008 
98906 i68856 
98902 | 169702 


i4 
14 
14 
1 4 
14 
14 
i4- 
U 
14 

u 


33 
3o 
27 
24 
22 
19 
16 
i3 
10 
07 


9-995372 
99(5353 
995334 
9903 i 6 
995297 
995278 
995260 
9952-41 
99.5222 
995203 


14-64 
14-61 
14-58 
14-55 
14-53 
i4-5o 

14-47 
14.44 

U-42 
U-39 


31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
'58 
59 
60 


14810 
14838 
14867 
14896 
14925 
14954 
14982 
i5oii 
i5o4o 
15069 
1 5097 
i5i26 
i5i55 
i5i84 

l52I2 

i534i 
15270 
15299 

i5327 
i5356 

15385 

i54i4 

15442 
1 547 1 
i55oo 
15529 
i5557 
i5586 
i56i5 
1 5643 


98897 
98893 
98889 
98884 
98880 
98876 
98871 
98867 
9 8863 
98808 


9.170547 
171389 
172230 
173070 
1 73908 
174744 
175578 
1 7641 1 
177242 
178072 


14 
14 

i3 

i3 
i3 
i3 

I«J 

r3 
i3 

i3 


o5 
02 

99 

96. 

94 
9i 

88 
86 
83 
80 


9-995184 
995 1 65 
995146 
995127 
995io8 
995089 
995070 
995o5i 
995o32 
995o 1 3 


-32 
-32 
-32 
-32 
•32 1 
•32| 
-32 
•32^ 
-32 
-32 


14-36 
14-33 
i4-3i 
1.4.28 
14-26 
14-23 
14-20 

14-17 
i4-i5 
14- 12 


to- 824638 
823776 
822916. 
822o58 
821201 
820345 
819492 
818640 
817789 
816941 


98854 
98849 
98845 
98841 
98836 
9883 a 
98827 
98823 
98818 
98814 


9-178900 
179726 
i8o55i 
181374 
182196 
i83oi6 
183834 
i8465i 
i85466 
1862-80 


i3 
e3 
i3 
i3 

i3 
i3 

1 3 
i3 

i3 

i3 


77 
74 
72 
69 
66 
64 
61 

59 
56 
53 


9.994993 
994974 
994955 
994935 
994916 
994896 

994877 
994857 
994838 
994818 


-32 
-32 
-32 
-32 1 
-33! 
-33 
-33 
-33 
-33 
-33 

~33~ 
-33 
•33 
•33 
•33 
•33 

!-33 
•33 
-33 
•33 


8 • 1 83907 
184752 
185597 
18643.9 
187280 
188120 
i88 9 58 
189794 
190629 
191462 


14-09 
24-07 
14-04 
14-02 
i3-99 
13.96 
13.93 
13-91 
i3-8& 
i3-86 


£0-816093 
816248 
8i44o3 
8i356i 
812720 
811880 
811042 
810206 
80987 i 
808 5 38 


98809 
9 88o5 
98800 
98796 
98791 
98787 
98782 
98178 
98773 
98769 


9.187092 
187903 
188712 
i8 9 5i 9 
19032O 
191130 
191 933 
192734 
193534 
194332 


i3 

1 3 
i3 
i3 
i3 
1 3 
i3 
i3 
i3 
i3 


Si 

48 
46 
43 

4T 

38 
36 
33 
3o 

28 


9.994798 

994779 
994759 
994739 
994719 
994700 
994680 
994660 
994640 
994620 


9- 192294 
193124 
1 93953 
194780 
195606 
1 96430 
197253 
1 98074 
198894 
1997 1 3 


i3-84 
i3-8i 
13.79 
13.76 
13-74 
18-71 
13-69, 
13-66 
13-64 
i3-6i 


to- 807706; 
806876 
806047 
80 5 220. 
804394 
803570 
802747 
801926 
801 106 
800287 


N. cos. j N. sine. 1 ~h. c©s. 


D.l" 


L. sine. 




L. cot. D.l" 


lu tang-. 


' i 


81° 



BIKES AND TANGENTS 



27 



Rad. = 1. 




Logarithms. — 


Radius = 10 10 . 




' 


N. sinej 1ST. cos. ! 

1 


L. sine. 


D. 1*' 


L. cos. 


D.l" 


L. tang. 


D.l" 


L. cot. 







1 5643 


98769 


9-I94332 


i3-28 


9-994620 


■33 


9-199713 


i3-6i 


10-800287 


60 


1 


15672 


98764 


196129 

195926 


13.26 


994600 


•33 


200529 


i3.5 9 


799471 


59 


2 


15701 


98760 


i3-23 


99458o 


• 33 


2oi340 


i3-56 


798655 


58 


8 


15730 


98755 


196719 


13-21 


99456o 


•34 


202159 


i3-54 


797841 


57 


4 


15758 


98751 


197011 


i3-i8 


994540 


• 34 


202971 


i3.52 


797029 


56 


5 


15787 98746 


198302 


i3-i6 


994519 


•34 


203782 


i3-49 


796218 


55 


6 


i58i6 98741 


1 9909 1 


i3-i3 


994499 


•34 


204592 


i3-47 


795408 


54 


7 


i5845 98737 


199879 


i3-u 


994479 


•34 


2o54oo 


13.45 


794600 


53 


8 


15873 I 98732 


200666 


i3-o8 


994459 


• 34 


206207 


13-42 


7 9 3 79 3 


52 


9 


15902 98728 


2oi45i 


i3-o6 


994438 


•34 


207013 


i3-4o 


792987 


51 


10 
11 


i5 9 3i ! 
15959 


98723 


202234 


i3-o4 


994418 


• 34 
•34 


207817 


i3-38 


792183 


50 
49" 


o8~7i8~ 


9-203017 


i3-oi 


9.994397 


9.208619 


13.35 


io-79i38i 


12 


15988 98714 


203797 


12-99 


994377 


•34 


209420 


i3-33 


790580 
789780 


48 


13 J 


16017 | 98709 


204577 


12-96 


99435] 


•34 


210220 


i3-3i 


47 


14 1 


16046 ! 98704 


2o5354 


12-94 


994336 


•34 


211018 


i3-28 


788982 


46 


15 | 16074 98700 


2o6i3i 


12-92 


9943i6 


• 34 


2ii8i5 


i3-26 


788185 


45 


16 i6io3 98695 


206906 


12-89 


994295 


•34 


212611 


i3-24 


787889 


44 


17 i6i32 98690 


207679 


12-87 


994274 


•35 


2i34o5 


13-21 


786595 


43 


18 


1 6 1 60 98686 


208452 


12-85 


994254 


•35 


214198 


i3 • 19 


785802 


42 


19 


16189' 98681 


209222 


12-82 


994233 


•35 


214989 


i3-i 7 
i3-i5 


785oi 1 


41 


20 

21 


16218 98676 
16246 98671 


209992 

9-210760 

211626 


12.80 


994212 


•35 
• 35 


215780 
9T2T6568 


784220 
f<T78343Y 


40 
~39~ 


12.78 


9-994191 


13-12 


22 


16275 98667 


12-75 


994171 


• 35 


217356 


i3-io 


782644 


38 


23i 


i63o4 98662 


212291 


12.73 


9941 5o 


• 35 


218142 


i3-o8 


781858 


37 


24! 


16333 98657 


2i3o55 


12.71 


994129 


•35 


218926 


i3-o5 


781074 


36 


25! 


i636i ! 9 8652 


2i38i8 


12-68 


994108 


• 35 


219710 


i3-o3 


780290 


35 


26! 


16390 98648 


214579 


12-66 


994087 


•35 


220492 


i3-oi 


779508 


34 


27 | 


16419 ', 98643 


2i5338 


12-64 


994066 


•35 


221272 


12-99 


778728 


33 


28 


16447 i 9^638 


216097 


12-61 


994045 


•35 


222052 


12.97 


777948 


32 


29! 


16476 '. 9 8633 


216804 


12-59 


994024 


• 35 


222830 


i2- 9 4 


777170 


31 


30 | 

31 


i65o5 I 98629 
i6533 ! 98624 


217609 
9^2i8363" 


12.57 


994003 


•35 


2236o6 


12-92 


776394 


30 


12-55 


9-993981 | -35 


9-224382 


12-90 


10-775618 29 


32' 


i6562 98619 


219116 


12-53 


993960 i -35 


225i56 


12-88 


774844 


28 


33 


16591 ' 98614 


219868 


12-50 


993939 1 -35 


225929 


12-86 


774071 


27 


34 


16620 ' 98609 


220618 


12-48 


993918 | -35 


226700 


12.84 


7733oo 


26 


35 


16648 98604 


221367 


12-46 


993896 | -36 


227471 


12-81 


772529 


25 


36 


16677 : 98600 


222110 


12-44 


993875 | -36 


228239 , 12-79 


771761 24 


37 


16706 98595 


222861 


12-42 


993854 


•36 


229007 ! 12-77 


770993 | 23 


38 


16734 98090 


2236o6 


12-39 


9 9 3832 


•36 


229773 ' 12-75 


770227 ! 22 


39 


! 16763 cy8585 


224349 


12.37 


99381 1 


•36 


23o539 | 12-73 


769461 | 21 


40 
IT 


16792 . 98580 
| 16820 98575 


225092 

9-225833 


12-35 


993789 


•36 
•36 


23i3o2 j 12-71 


768698 j 20 


12-33 


9-993768 


9-232o65 1 12-69 


10-767935 |19 


42 


! 16849 '98570 


226573 


12-31 


993746 i -36 


: 232826 ! 12-67 


767174 18 


43 ;j 16878 


98565 


22731 1 


12-28 


993725 ! -36 


I 233586 12-65 


766414 17 


44 


j 16906 


9 856i 


228048 


12-26 


993703 1 -36 


234345 12-62 


765655 16 


45 


! 16935 


98556 


228784 


: 12-24 


99368i .36 


1 235io3 12-60 


764897 1 15 


46 


! 16964 


g855i 


229518 


j 12-22 


99 366o .36 


23585 9 | 12-58 


764141 |14 


47 


16992 


98546 


230252 


' 12-20 


993638 j -36 


i 2366i4 ! 12-56 


763386 13 


48 


17021 


98541 


230984 


i 12.18 


9936i6 -36 


! 237368 12-54 


762632 12 


49 


i7o5o 


9 8536 


23i7i5 


! 12-16 


99 35 9 4 .37 


238120 12-52 


761880 ill 


50 
51 


i_70?8 
17107 


9853i 
98326"" 


232444 
9-233172 


! 12-14 
1 12-12 


99 35 7 2 i -3 7 
9-99355o j .37 


238872! 12 -5o 


761 1 28 10 


9-239622 j 12-48 


10-760378 9 


52 


17136 


98521 


233899 


| 12-09 


993528 .37 


1 240371 j 12-46 


7596291 8 


53 


17164 


985i6 


234620 


j 12-07 


9935o6 j .37 


241118 


12-44 


7 58882 j 7 


54 


17193 


985i 1 


235349 


j 12 -o5 


993484 j - 37 


241860 


12-42 


758i35 6 


55 


17222 


9 85o6 


236073 


12 -o3 


993462 .37 


242610 


12-40 


7573901 5 


56 


i725o 


98001 


236795 


! 12-01 


993440 j .37 


243354 


12-38 


756646 1 4 


57 


17279 


98496 


23751 5 


j n-99 


993418 j -3 7 


244097 


12-36 


755903 3 


58 


! 17308 


9S491 


238235 


11.97 


993396 j .37 


244839 


12-34 


755i6i 2 


59 


! i 7 336 


98486 


238953 


11-95 


993374 .37 


245579 


12-32 


754421 1 


60 


j 17365 


98481 


239670 


11-93 


99335i | .37 


2463i 9 


i2-3o 


75368i 1 


N. cos. N. sine. 


| L. cos. 


D. 1" 


L. sine. 1 


L. cot. 


D. 1" 


L. tang. 


80° 












11* 













28 



SINES AND TANGENTS. 10 c 



Rad. = 1. 






Logarithms. — Radius = 


= 10'°. 







N. sine. 
i 7 365 


N. cos. 


L. sine. 


D.l" 


D. cos. 


D.l" 


L. tang. 


D. 1" 


L. cot. 




98481 


9-239670 


11-93 


9.9933D1 


•3 7 


'9-246319 


12-30 


io-75368i 


60 


1 


17393 98476 


24o386 


11. 91 


993329 


•3 7 


247057 


12.28 


762943 


59 


2 


17422 j 98471 


241101 


n-89 


993307 


•3 7 


247794 
24853o 


12-26 


762206 


58 


3 


17461 1 98466 


241814 


11-87 


993285 


.37 


12-24 


701470 


57 


4 


17479 98461 


242526 


n-85 


993262 


.37 


249264 


12-22 


760736 


56 


5 


17308 98455 


243237 


n-83 


993240 


•3 7 


249998 


12-20 


730002 


55 


6 


17537 98450 


243947 


11-81 


993217 


• 38 


260730 


I2-I8 


749270 


54 


7 


17565 98445 


244656 


n-79 


993195 


• 38 


201461 


12-17 


748539 


53 


S 


17394 


98440 


245363 


11.77 


993172 


• 38 


2D2 191 


12- 15 


747809 


52 


9 


17623 


98435 


246069 


ii. 7 5 


993149 


• 38 


252920 


I2-I3 


747080 


51 


10 
ll' 


17661 98430 


246775 
9-247478 


11.73 


99 3l2 7 
9-993104 


•38 
"38 


253648 


12-11 


746352 
10-745626 


50 
49 


1 7680 


98425 


11. 71 


9.254374 


12-09 


12 


17708 


98420 


248181 


11-69 


993081 


•38 


255ioo 


12-07 
I2-o5 


744900 


48 


13 


17737 


984U 


248883 


11-67 


993069 


•38 


255824 


744176 


47 


14 


17766 


98409 


249583 


H-65 


993o36 


• 38 


256547 


12-03 


743453 


46 


15 


'7794 


98404 


250282 


n-63 


993oi3 


• 38 


257269 


12-01 


742731 


45 


16 


17823 


98399 


250980 


n-6i 


992990 


• 38 


267990 
268710 


12-00 


742010 


44 


17 


17802 


9S394 


251677 


11.59 


992967 


• 38 


II.98 


741290 


43 


18 


17880 


9 838 9 


252373 


u-58 


992944 


• 38 


269429 


II.96 


740571 


42 


19 


17909 


9 8383 


253067 


11-56 


992921 


• 38 


260146 


u-94 


739864 


41 


20 

ii" 


17937 
17966 


98378 


253761 


u-54 


992898 


• 38 
•38 


26o863 


11.92 


739i37 

10-738422 


40 
39 


9837T 


9-254453 


11-32 


9.992875 


9-261578 


11-90 


22 


17990 


98368 


255i44 


II -30 


992832 


• 38 


262292 


11-89 


737708 


38 


23 


18023 


98362 


255834 


11.48 


992829 


•3 9 


263oo5 


11.87 


736995 
736283 


37 


24 


i8o52 


98357 


256523 


11-46 


992S06 


• 3 9 


263717 


u-85 


36 


25 


18081 


98352 


257211 


H--44 


992783 


• 3 9 


264428 


n-83 


735372 


35 


26 


18109 


93347 


257898 


11.42 


992739 


• 3 9 


265i38 


ii-8i 


734862 


34 


27 


i8i38 


9 834i 


258583 


11. 41 


992736 


•3 9 


265847 


11.70 


734i53 


33 


28 


18 166 


98336 


259268 


11.39 


992713 


• 39 


266555 


11.78 


733445 


32 


29 


i8i 9 5 


9833i 


259951 


11.37 


992690 


•3 9 


267261 


11.76 


732739 


31 


30 
31 


18224 
18262 


98325 


26o633 


u-35 


992666 
9.992643 


• 3 9 

• 3 9 


267967 


u-74 


732o33 


30 
29 


98320 


9'26i3~i4 


n-33 


9-268671 


11.72 


10-731329 


32 


18281 


9 83i5 


261994 


11 -3i 


992619 


1° 


269375 


11.70 


73o623 


28 


33 


1 83 09 
i8338 


98310 


262673 


u-3o 


992596 


■3.9 


270077 


11.69 


729923 


27 


34 


98304 


26335i 


11.28 


992572 


• 3 9 


270779 


11.67 


729221 


26 


35 


i836 7 


98299 


264027 


11.26 


992549 


•3 9 


271479 

272178 


n-65 


728521 


25 


36 


18395 


98294 


264703 


11-24 


992525 


• 3 9 


11-64 


727822 


24 


37 


18424 


98288 


265377 


11-22 


992501 


•3 9 


272876 


11-62 


727124 


23 


38 


18452 


98283 


266o5i 


11-20 


992478 


• 40 


273573 


u-6o 


726427 


22 


39 


1 8481 


98277 


266723 


II- I 9 


992454 


.40 


274269 


n-58 


725731 


21 


40 
41 


i85o 9 

i8538 


98272 
98267 


267395 
9-268065 


11-17 


992430 


• 40 


274964 


11.57 


725o36 


20 


11. 15 


9-992406 


.40 


9-275658 


u-55 


10-724342 


19 


42 


1 8567 


98261 


268734 


u-i3 


992382 


• 40 


276351 


n-53 


723649 


18 


43 


18095 


98266 


269402 


II -II 


992359 


.40 


277043 


ii-5i 


722957 


17 


44 


18624 


98250 


270069 


11-10 


992335 


•40 


277734 


n-5o 


722266 


16 


45 


18662 


98245 


270730 


IT* 68 


9923 1 1 


• 40 


278424 


u-48 


721676 


15 


46 


18681 


98240 


271400 


11-06 


992287 


•40 


2791 i3 


n-47 


720887 


14 


47 


18710 


98234 


272064 


n-o5 


992263 


• 40 


279801 


u-45 


720199 


13 


48 


18738 


98229 


272726 


11-63 


992239 


.40 


280488 


n-43 


719512 


12 


49 


18767 


98223 


2 7 3388 


II -01 


992214 


• 40 


281 174 


11.41 


718826 


11 


50 
51 


i8 79 5 
18824 


98218 


274049 


10-99 


992190 


•40 


28i858 


ii-4o 


718142 


10 


98212 


9.274708 


10-98 


9.992166 


.40 


9.282342 


n-38 


10-717468 


9 


52 


i8852 


98207 


275367 


10-96 


902142 


.40 


283225 


n-36 


716775 


8 


53 


1 888 1 


98201 


276024 


10-94 


992117 


•4i 


283907 
284588 


u-35 


716093 


7 


54 


18910 


98196 


276681 


10-92 


992093 


•4i 


n-33 


716412 


6 


55 


i8 9 38 


98190 


2 77 33 7 


10-91 


992069 


•4i 


285268 


11. 3i 


714732 


5 


5(5 


18967 


98185 


277991 


10.89 


992044 


•41 


286947 


n--3o 


7i4o53 


4 


57 


i8 99 5 


98179 


278644 


10-87 


992020 


•4i 


286624 


11-28 


713376 


3 


58 


18024 


98174 


279297 


10-86 


991996 


•4i 


287301 


11-26 


712699 


2 


59 


i8o52 


98168 


279948 
280699 


10-84 


991971 


•41 


287977 


11-25 


712023 


1 


60 


18081 


9 8i63 


10-82 


991947 


•41 


288652 


11-23 


711348 





N. cos. N. sine. 


L. cos. 


D.l" 


L. sine. 




L. cot. 


m \ 


L. tang. 


/ 


79 o 



SINES AND TANGENTS. 11°. 



29 



Rad. = 1. 


1 


Logarithms.— 


-£adius = 10 10 . 




i ' 


N.sine. N. cos. 


L. sine. 


D. 1" L. cos. 


D.l" 


L. tang. D. \" 


L. co c. 




| 


19081 ' 9 8i63 


1 9-280699 


10-82 


| 9.991947 


•41 


9-288652 n- 23 


10-711348 


60 


1 [j iqi09 9S 1 57 


281248 


iq.8i 


991922 


•4i 


289326! 11-22 


710674 


59 


2 ; j igi38; 9816:2 


281897 


10-79 


991897 


.41 


289999 j 11-20 


710001 


58 


S !| 19167 j 98146 


282544 


10-77 


991873 


•4i 


290671 


11-18 


709329 


57 


: 4 


19190 | 98*40 


223190 


10-76 


991848 


•4i 


291342 


11-17 


708668 


56 


1 5 


19224! 98135 


283836 


10-74 


991823 


■41 


292013 


1 1 • i5 


707987 


55 


6 


19252 98139 


284480 


10-72 


99*799 


•41 


292682 


ii-i4 


707318 


54 


: 7 


19281 


98124 


286124 


10-71 


99*774 


•42 


293350 


11-12 


7o665o 


53 


! 8 


19309 


98118 


286766 


10-69 


Q91749 


•42 


294017 


11-n 


705983 
7o53i6 


52 


• 9 


19338 


981 1 2 


286408 


10-67 


99 1 7 24 


•42 


294684 


11-09 


51 


! 10 


i 9 366 
59390 


98107 
98101 


287048 
9^87687 


10-66 


99 l6 99 
9.991674 


•42 

•42 


295349 
9-296013 


11-07 


704651 


50 
49 


10-64 


ii -06 


10-703987 


12 


19423 


98096 


288326 


io-63 


991649 


•42 


296677 


11 -04 


703323 


48 


13 


19402 


98090 


288964 


io-6i 


991624 


.42 


297339 


n-o3 


702661 


47 


14 


1 948 1 


98084 


289600 


1 *. $0 


991699 


•42 


2980OI 


II-OI 


701999 


46 


IS 


19009 


98079 


290236 


io-58 


991674 


•42 


298662 


11-00 


70i338 


45 


16 


19038 


98073 


290870 


jo-56 


991549 


•42 


299322 


10-98 


700678 


44 


17 


19066 


98067 


291604 


10-54 


991624 


•42 


299980 


10-96 


700020 


43 


18 


19696 


98061 


292137 


io-53 


99U98 


•42 


3oo638 


10-96 


699362 


42 


19 


19623 


98066 


292768 


io-5t 


99U73 


•42 


301295 


10-93 


698705 


41 


20 
"21 


19662 


98060 


293399 


io-5o 


991448 


•42 
•42 


301961 

9.302607 


10-92 


698049 


40 


i"^68o 


98044" 


9-294029 


10-48 


9-991422 


10-90 


10-697393 


39 


22 


19709 


98039 


294668 


10-46 


991397 


•42 


3o326i 


10-89 


696739 


38 


23 


19737 


98o33 


296286 


10-45 


991372 


.43 


303914 
304067 
3o52i8 


10-87 


696086 


37 


24 


19766 


98027 


296913 


10-43 


991346 


•43 


io-86 


695433 


36 


25 


19794 


98021 


296539 


io- 42 


90 1 32 1 


•43 


10-84 


694782 


35 


26 


1.983.3 


98016 


297164 


10-40 


991296 


•43 


3o586 9 


io-83 


6941 3 1 


34 


27 


19861 


98010 


297788' 


10-39 


991270 


•43 


3o65i9 


10.81 


693481 


33 


28 


19880 


98004 


298412 


10-37 


991244 


-43 


307168 


io-8o 


692832 


32 


29 


19908 


97998 


299034 


io-36 


9912 1 8 


•43 


307815 


10-78 


692185 


31 


30 
31 


*99 3 7 

19966 


97992 


299665 
9-300276 


io-34 

10-32 


991 193 
9-991167 


-43 
•43 


3o8463 


10-77 


691637 


30 


97987 


9*309109 


10-76 


10-690891 


29 


32 


10994 


9798i 


300895 


io-3i 


99 "4i 


•43 


309764 


10-74 


690246 
689602 


28 


33 


200-2 2 


97975 


3oi5i4 


10-29 


991115 


•43 


310398 


19.73 


27 


34 


2oo5i 


97969 


302132 


10-28 


991090 


•43 


3uo42 


10-71 


688 9 58 


26 


35 


20079 


97963 


302748 


10-26 


991064 


•43 


3u685 


10-70 


6883i 5 


25 


36 


2010(8 


97908 


3o3364 


10-25 


99io38 


•43 


3i2327 


10-68 


687673 


24 


37 


2oi36 


9 79 52 


303979 


10-23 


991012 


•43 


312967 


10-67 


687033 


23 


3S 


20166 


97946 


304093 


10-22 


990986 


•43 


3i36o8 


io-65 


6863 9 2 


22 


39 


20193 


97940 


300207 


10-20 


990960 


•43 


314247 


10-64 


685 7 53 


21 


40 


20222 


97934 
97928 


3o58i9 
9-3o643o 


io- 19 


990934 


•44 

•44 


3i4885 
9-3i5523 


10-62 


685ii5 


20 


41 1 20260 


io- 17 


9-990908 


io-6i 


10-684477 


19 


42 |j 20279 


97932 


307041 


10-16 


990882 


•44 


316169 


io-6o 


683841 


18 


43 20307 


97916 


307660 


10-14 


990855 


•44 


316796 


10-58 


683205 


17 


44 j! 20336 


97910 


3o825 9 


io- 13 


990829 


•44 


3i743o 


10-57 


682670 


16 


45 20364 


97906 


308867 


IO-II 


990803 


•44 


3i8o64 


io-55 


681936 


15 


46 ! 2o3g3 


97899 


309474 


IO-IO 


990777 


•44 


318697 


io-54 


68i3o3 


14 


47 ! 20421 


97893 


3 1 0080 


10-08 


990750 


•44 


319329 


io-53 


680671 


13 


48 2o45o 97887 


3 10686 


10.07 


990724 


•44 


319961 


io-5i 


680039 
679408 


12 


49 |j 20478 97881 


3i 1289 


10 -o5 


990697 


•44 


320692 


io-5o 


11 


50 


20507 | 97870 


3u8 9 3 


10-04 


990671 


•44 


321222 


10-48 


678778 


10 


51 


2o535 97869 


9-312496 


10 -o3 


9-990644 


•44 


9-32i85i 


10 -47 


10-678149 


9 


52 20563 | 97863 j 


313097 


10-01 


990618 


•44 


322479 


io-45 


677621 


8 


53 | 20092 j 97867 


3i36 9 8 


10-00 


990691 


•44 


323io6 


io-44 


676894 


7 


54 j 20620 97861 


314297 


9-98 


99o565 


•44 


323 7 33 


io-43 


676267 


6 


55 i 20649 i 97845 


314897 


9.97 


990538 


•44 


324358 


10-41 


676642 


5 


56 ! 20677 97839 


316496 


9.96 


99o5n 


•45 


324983 


lo-4o 


675017 


4 


57 |! 20706! 97833 


316092 


9.94 


990485 


•45 


325607 


10-39 


674393 


3 


58 : . 20734 97827 


3i668 9 


9 - 9 3 


990468 


•45 


32623i 


10-37 


673769 


2 


59 20763 97821 


317284 


9-91 


990431 


•45 


326853 


io-36 


673147 


1 


60 j 20791 97815 


317879 


9.90 


990404 


•45 


327475 


io-35 


672525 





[N. cos. N. sine. 


L. cos. 


D.l" 


L. sine. 




L. cot. 


"5^- 


L. tang. 






78° 









30 



SINES AND TANGENTS. 12°. 



Rad. = 1. 






Logarithms. — 


Radius- 10 10 . 







N.sine. Ncos. 


L. sine. 


D. 1" 


L. cos. 


D.1" 


L. tang. 


D. 


1" 


L. cot. 




20791 


97815 


9-317879 
3i8473 


9 


•90 


9-990404 


•45 


9-327474 
328095 


IO 


35 


10-672526 


60 


1 


20820 97809 


9 


88 


990378 


•45 


10 


33 


671905 


59 


2 


20848 97 8o3 


319066 


9 


.87 


9903 5 1 


•45 


328 7 i5 


10 


32 


671285 


58 


3 


20877 


97797 


3i9658 


9 


-86 


990324 


•45 


329334 


10 


3o 


670666 


57 


4 


20905 


97791 


320249 


9 


-84 


990297 


•45 


329953 


10 


3 


670047 


56 


5 


20933 


97784 


320840 


9 


• 83 


990270 


•45 


33o57o 


10 


669430 


55 


6 


20962 


97778 


32i43o 


9 


82 


990243 


•45 


33ii8 7 


10 


26 


6688 1 3 


54 


7 


20990 


97772 


322019 


9 


80 


990215 


•45 


33i8o3 


10 


25 


668197 


53 


8 


21019 


97766 


322607 


9 


•79 


990188 


-45 


3324i8 


10 


24 


667582 


52 


9 


21047 


97760 


323194 


9 


77 


990161 


•45 


333o33 


10 


23 


666967 


51 


10 
U 


21076 
21 104 


97754 


323780 


9 


76 


990 1 34 


•45 


333646 
9-334259 


10 


21 


666354 


50 
49 


97748 


9-324366 


9 


7 5 


9-990107 


-46 


10 


20 


10-665741 


12 


2Il32 


97742 


324950 


9 


73 


990079 


-46 


334871 


10 


}9 


665 1 29 


48 


13 


2Il6l 


97735 


325534 


9 


72 


990052 


•46 


335482 


10 


17 


6645i8 


47 


14 


21 189 
2I2I8 


97729 


326i 17 


9 


70 


990025 


•46 


336093 


10 


16 


668907 


46 


15 


97723 


326700 


9 


69 


989997 


•46 


336702 


10 


i5 


6632 9 8 


45 


16 


21246 


97717 


327281 


9 


68 


989970 


• 46 


3373 1 1 


10 


i3 


662689 


44 


17 


21275 


97711 


327862 


9 


66 


989942 


-46 


337919 


10 


12 


662081 


43 


18 


2i3o3 


97700 


328442 


9 


65 


989915 


-46 


338527 


10 


1 1 


66i473 


42 


19 


2i33i 


97698 


329021 


9 


64 


989887 


•46 


33 9 i 33 


10 


10 


660867 


41 


20 
~21~ 


2i36o 

2i388 


97692 
97686 


329599 
9-33on6 


9 


62 


989860 


-46 
-46 


339789 


10 


08 


660261 


40 
39 


9 


61 


9-989832 


9 • 34o344 


10 


07 


io-65o656 


22 


214H 


97680 


330753 


9 


60 


989804 


-46 


340948 


10 


06 


65qo52 


38 


23 


21445 


97673 


33i329 
33i9o3 


9 


58 


989777 


.46 


34i552 


10 


04 


658448 


:37 


24 


21474 


97667 


9 


5 7 


989749 


•47 


342i56 


10 


o3 


657845 


36 


25 


2I&02 


97661 


332478 


9 


56 


989721 


•47 


342757 


10 


02 


657243 


35 


26 


2i53o 


97655 


333o5i 


9 


54 


989693 


•47 


343358 


10 


00 


606642 


34 


27 


2i559 


97648 


333624 


9 


53 


989665 


•47 


343958 


9 


99 


656o42 


33 


28 


2i58 7 


97^42 


334195 


9 


52 


989637 


•47 


344558 


9 


98 


655442 


32 


29 


21616 


97636 


334766 


9 


5o 


989609 


•47 


345 1 57 


9 


97 


654843 


31 


30 
~3T 


21644 


97630 


335337 


9 


49 


989582 


•47 


345755 


__2_ 


i^_ 


654245 


30 
~29~ 


21672 


97623 


9-333906 


9 


48 


9-989553 


•47 


9.346353" 


9 


94 


io-653647 


32 


21701 


97617 


336475 


9 


46 


989525 


•47 


346949 


9 


9 3 


653o5i 


28 


33 


21729 


9761 1 


337043 


9 


45 


989497 


•47 


347545 


9 


92 


652455 


27 


84 


21758 


97604 


33 7 6io 


9 


44 


989469 


•47 


348141 


9 


91 


65i85 9 


26 


35 


21786 


97598 


338176 


9 


43 


989441 


•47 


348735 


9 


90 


66)265 


25 


36 


21814 


97592 


338742 


9 


41 


9 8 9 4i3 


•47 


349329 


9 


88 


650671 


24 


37 


2i843 


97585 


33g3o6 


9 


40 


989384 


•47 


349922 


9 


87 


650078 


23 


38 


21871 


97579 


339871 


9 


39 


989356 


•47 


35o5i4 


9 


86 


649486 


22 j 


39 


21899 


97573 


340434 


9 


37 


989328 


•47 


35no6 


9 


85 


648894 


21 I 


40 


21928 
219D6 


j97_566 

j?756o 


340996 
9-34x558 


9 
9 


36 
35 


989300 
9-989271 


•47 
•47 


351697 


_9^ 


83 
82" 


6483o3 
i • 6477 1 3 




9-352287 


9 


19 1 


42 


21985 


97553 


342119 


9 


34 


989243 


•47 


352876[ 9 


81 


647124 


18 


43 


220l3 


97547 


342679 


9 


32 


989214 


•47 


353465 j 9 


80 


646535 17 j 


44 


22041 


97^4i 


343239 


9 


3i 


989186 


•47 


354o53 9 


79 


645947 16 ' 


45 


22070 


97534 


343797 


9 


3o 


989/57 


•47 


354640 9 


77 


645360 ! 1 5 


46 


220Q8 


97528 


344355 


9 


29 


989128 


-48 


355227 9 


76 


644773 1* 


47 


22 126 


97521 


344912 


9 


27 


989 1 00 


-48 


3558i3 ' 9 


7 D 


644187 ' 13 


48 


; 22 I 55 


975 1 5 


345469 


9 


26 


989071 


•48 


356398 9 


"4 


643602 12 ; 


1 49 


22183 


975oS 


346024 


9 


25 


989042 


■48 


356982 ; 9 


73 


643oi 8 11 | 


50 
j 5T 


22212 


97502 


3465 79 


9 


24 


989014 | 

9 -9889851' 


•48 

"•78 


35/566 
9 -3 58 149 


9 


71 


__6_42434 




222 40 97496 
22268 j 97489 


9-347i34 


9 


22 


9 


"TO 


10 -6418,51 


"9 : 


52 


347687 


9 


21 


9 88 9 56 1 


-48 


358 7 3. 9 


69 


64 1 269 


8 


1 ^ 


22297 97483 


348240 


9 


20 


988027 ! 


•48 


3593i3 9 


fStS 


640687 


7 


\ ^ 22325 C/H-b 


348702 


9 


'9 


988898 


.48 


35989.3 9 • 


6: M07 


"5 22353 q- 170 


340343 


9 


'7 


9 8886 9 


•48 


3(x>^74 < 




j 56 22382 j 9^463 


349^3 


9 


16 


988840 


•48 


36io53 9 


638947 4' 


j 57 


22410 


97457 


35o443 


9 


i5 


98881 1 1 


.49 


36i632 9-63 


638368 




1 5S 


22438 


9745o 


350992 


9 


14 


988782 1 


•49 


362210 9-62 


637790 


2 


1 59 


22467 


97444 


35i54o 


9 


i3 


988753 | 


.49 


36*787 


9.61 


6372i3 


1 


1 6 ° 


22495 


97437 


352088 


9. u 


988724 


•4 9 


363364 


9-60 


636636 


j 


f 


N.cos. N. sine. 


L. cos. 


D.l" 


L. sine. 




L. cot. 


D. V 


L. tang. 


~{ 


f^° 



SINES AND TANGENTS. 13°. 



31 



Rad. = 1. 




Logarithms. — Radius = 10 10 . 


...... : 


' N. sine. N. cos. | 


L. sine. 


D. 1" 


L. cos. 1 


M "| 


L. tang. 


Dl." | 


L. cot. 




of 


22495 1 97437 


9-352o88 


9. 11 


9-988724 


•49 


9-363364 


9-60 


to- 636636 60 


1 | 22023 


9743o 


352635 


9-10 


988695 


•49 


363940 


9 -5 9 


636o6o|59 


2 i 22552 ! 


97424 


353i8i 


9.09 


988666 


-49 


3645i5 


9-58 


635485 58 


S 2258o t 


97417 


353726 


9-08 


9 88636 


•49 


365090 


9-^7 


634910 57 ; 


4|j 226o8 ; 


9741 1 


354271 


9-07 


988607 


•49 


365664 


9-5o 


634336 56 


5 II 22637 | 


97404 


3548i5 


9>o5 


988578 


•49 


366237 


9-54 


633763I 55 


6 ! 2 2665 


97398 


355358 


9-04 


988548 


•49 


3668io 


9-53 


633190 


54 


7 i, 22693 


97391 


355901 


9«o3 


988519 


-49 


36 7 382 


9 .52 


6326i8 


53 


8 22722 


97384 


356443 


9-02 


988489 


•49 


367953 


9«5i 


632047 


52 


9 i 22750 


97378 


356984 


9-01 


988460 


•49 


368524 


9'5o 


63i476 


51 


10 
11 


22778 
22807 


9737J J 
97365 


357524 


8-99 


988430 


.49 


369094 
9-369663 


9.49 


680906 


50 , 
49 


9- 358o64 


8-98 


9.988401 


-49 


9-48 


io-63o337 


12 


22835 


97 358 


3586o3 


8.97 


9 883 7 i 


-49 


370232 


9.46 


629768 


48 


13 


22863 


9735i 


359141 


8-96 


988342 


•49 


370799 


9-45 


629201 


47 ■ 


14 


22892 


97345 


359678 


8- 9 5 


9 883 1 2 


-5o 


371367 


9.44 


628633 


46 : 


15 


22920 


97338 


36o2i5 


8- 9 3 


988282 


-5o 


371933 


9-43 


628067 


45 


16 


22948 


9733i 


360752 


8-92 


988252 


• 5o 


372499 


9-42 


627501 


44 


17 


22977 


97325 


361287 


8-91 


988223 


• 5o 


373064 


9-4i 


626986 


43 


18 


23oo5 


97318 


361822 


8-90 


988193 


• 5o 


373629 


9.40 


626371 


42 


19 


23o33 


973 u 


362356 


8-89 

8-88 

~8^8 7 


988163 


• 5o 


374193 


9.39 


620807 


41 ! 


20 

|21 


23o62 


97304 


362889 


988i33 
9 • 988 1 o3 


-5o 
• 5o 


374756 


9-38 


620244 


40 | 

"39 ! 


23090 


97298 


9-363422 


9-375319 


9.37 


10-624681 


1 22 


23n8 


97291 


363954 


8-85 


988073 


• 5o 


37588i 


9-35 


6241 19 


38 | 
37 


| 2 : ' 


23i46 


97284 


364485 


8-84 


988043 


-5o 


376442 


9-34 


623558 


2-1 


28175 


97278 


365oi6 


8-83 


9 88oi3 


• 5o 


377003 


9-33 


622997 


36 i 


25 


23203 


97271 


365546 


8-82 


987983 


• 5o 


377563 


9-32 


622437 


35 i 


26 


2323l 


97264 


366075 


8-8! 


987953 


-5o 


378122 


9-3i 


621878 


34 


27 


23260 


97207 


3666o4 


8-8o 


987922 


-5o 


378681 


9-3o 


621319 


33 


28 


23288 


97201 


36 7 i3i 


8.79 


9S7892 


• 5o 


379239 


9-29 


620761 


32 : 


29 


233 16 


97244 


367659 


8-77 


987862 


•DO 


379797 


9-28 


620203 


31 : 


30 
31 


23345 

233 7 3 


97237 


368i85 


8-76 


987832 


-5i 


38o354 


9 •??'■■ 
9-26 


619646 


30 | 
~29~ ! 


97230 


9-368711 


8-70 


9-987801 


-5i 


9-380910 


10-619090 


32 


23401 


97223 


j 369236 


8-74 


987771 


-5i 


381466 


9>25 


6i8534 


28 . 


33 


23429 


97217 


369761 


8- 7 3 


987740 


-5i 


382020 


9-24 


617980 


27 


34 


23458 


97210 


370285 


8.72 


987710 


• 5i 


382575 


9-23 


617425 


26 


35 


23486 


97203 


370808 


8.71 


987679 


-5i 


383 1 29 


9-22 


616871 


25 


36. 


235?4 


97196 


37i33o 


8-70 


987649 


-5i 


383682 


9-21 


6i63i8 


24 


37 


23542 


97189 


371802 


8-69 


987618 


-5i 


384234 


9- 20 


615766 


23 


38 


23571 


97182 


372373 


8-67 


987O88 


-5i 


384786 


9.19 


6i52i4 


22 : 


39 


23599 


97176 


372894 


8-66 


987557 


■ 5i 


385337 


9-18 


6i4663 


21 


40 
III 


23627 
23656 


97169 
97162 


373414 
1 9-373933 


8-65 


987526 
9-987496 


-5i 
^57 


385888 


9-17 


614112 


20 


8-64 


9-386438 


9'i5 


io-6i3562 


W\ 


1 42 


23684 


97i55 


374452 


8-63 


987465 


-5i 


886987 


9-14 


6i3oi3 


18 


43 


23712 


97U8 


374970 


8-62 


987434 


•5i 


38t536 


9-i3 


612464 


17 


44 


23740 


97141 


375487 


8-6i 


987403 


•52 


388o84 


9-12 


611916 


16 


1 45 


23769 


97134 


376003 


8-6o 


987372 


-52 


38863i 


9-n 


61 i36g 


15 


46 


23797 


97127 


376519 


8-5 9 


987341 


•52 


389178 


9- 10 


610822 


14 


47 


23823 


97120 


377035 


8-58 


987310 


•52 


389724 


9-09 


610276 


13 


48 


23853 


97113 


377049 


8-57 


987279 
987248 


-52 


390270 


9-08 


609730 


12 


49 


23882 


97106 


j 378063 


8-56 


•52 


890815 


9.07 


609185 


11 


50 
ni 


23910 


91100 


1 • 378577 


8-54 


987217 


-52 


391360 


9-06 


608640 


10 


23938 


97093 


[9-379089 


8-53 


9-987186 


•52 


9-891903 


9-o5 


10-608097 


52 


28966 


97086 


j 379601 


8-52 


987155 


•52 


392447 


9.04 


607553 


8 


53 


1 23q95 


97079 


38ou3 


8-5i 


987 1 24 


•02 


392989 


9-o3 


60701 1 


7 


54 i 24028 


970-2 


380624 


8-5o 


987092 


•52 


393531 


9-02 


606469 


6 


55 24011 


91065 


38n34 


8-49 


987061 


•52 


394073 


9-01 


60:1927 


5 


; 56 24079 


97o58 


38i643 


8-48 


987030 


•52 


394614 


9-00 


6o5386 


4 


jj 57 1 24108 


i 97051 


382i52 


8-47 


986998 


•52 


390 1 54 


8-Q9 


604846 


3 


j 58 | 24 1 36 


' 97044 


382661 


8-46 


986967 


•52 


390694 


8-98 


6o43 06 


2 


| 59 i 24164 


97037 


383 168 


8-45 


9 86 9 36 


•52 


396233 


8.97 


603767 


1 


60 | 24192 


97o3o 


3836 7 5 


8-44 


986904 


•52 


396771 


8-96 


603229 





|jN. cos 


IN. sine 


. | L. cos. 


D.l" 


L. sine. 




L. cot. 


i D.l" 


L. tan?- 


, 


L 






73° 




I 



no 



SINES AND TANGENTS. 14°. 



Rad. = 1. 






Logarithms.— 


Radius — 


10'°. 




' 


N. sine. N. cos. 

| 


L. sine. 


D. 1'' 


L. cos. 


D.l" 


L. tang. 


D. 1" 


L. cot. 







24192 \ 97030 


19-383675 


8-44 


9-986904 


-52 


9.396771 


8-96 


10-603229 


60 


1 li 24220 97023 


i 384i82 


8 


43 


986873 


•53 


397309 


8-96 


602691 


59 


2 I) 24249 ' 97015 


i 384687 


8 


42 


986841 


•53 


397846 


8- 9 5 


602104 


58 


S 


1 24277 


97008 


! 385i92 


8 


4i 


986809 


•53 


3 9 8383 


8.94 


601617 


57 


4 


243o5 


97001 


1 383697 


8 


4o 


986778 


•53 


398919 


8- 9 3 


601081 


56 


1 5 


! 24333 


96994 


386201 


8 


3 9 


986746 


•53 


399455 


8-92 


6oo545 


55 


I 6 


1 24362 


96987 


386704 


8 


38 


986714 


-53 


399990 


8-91 


600010 


54 


7 


24390 


96980 


387207 


8 


i 1 


9 86683 


•53 


400324 


8-90 


599476 
598942 


53 


1 8 


i 24418 


96973 


387709 


8 


36 


9 8665i 


•53 


4oio58 


8-89 


52 


1 9 


24446 


96966 


388210 


8 


35 


986619 


•53 


401591 


8-88 


698409 


51 


10 


_24474 
245o3 


96959 
96962 


3887 1 1 


8 


34 


986587 


•53 
•53" 


402124 


8-87 


597876 50 


PIT 


9-389211 


8 


33 


9-986555 


9-402656 


8-86 


10-597344 49 


12 


2453i 


96945 


389711 


8 


32 


986523 


•53 


403187 


8-85 


5 9 68 1 3 


48 


13 


245D9 


96937 


390210 


8 


3i 


986491 


•53 


4o3 7 i8 


8-84 


596282 


47 


14 


24587 


96930 


390708 


8 


3o 


986459 


•53 


404249 

404778 


8-83 


595751 


46 


15 


24616 


96923 


391206 


8 


28 


986427 


• 53 


8-82 


595222 


45 


16 


24644 


96916 


391703 


8 


27 


986395 


•53 


4o53o8 


8-8i 


594692 


44 


17 


24672 


96909 


392199 


8 


26 


986363 


•54 


4o5836 


8-8o 


594164 


43 


18 


24700 


96902 


392695 


8 


25 


98633i 


•54 


4o6364 


8-79 
8-78 


5 9 3636 


42 


19 


24728 


96894 


3o3 191 


8 


24 


986299 


•54 


406892 


593108 


41 


20 

2T 


24756 
24784" 


96887 
96880" 


3 9 3685 


8 


23 


986266 


•54 


407419 
9-407945 


8.77 


592581 
io-592o55 


40 


9-394179 


8 


22 


9-986234 


•54 


8-76 


22 


248i3 


96873 


394673 


8 


21 


986202 


•54 


408471 


8-75 


591529 
591003 


38 


23 


24841 


96866 


3 9 5i66 


8 


20 


986169 


•54 


408997 


8-74 


37 


24 


24869 


9 6858 


395658 


8 


11 


986137 


•54 


409321 


8-74 


590479 


36 


25 


24897 


9 685i 


396130 


8 


986104 


•54 


410045 


8- 7 3 


58 99 55 


35 


2fi 


24925 


96844 


396641 


8 


n 


986072 


•54 


410569 


8-72 


58 9 43 1 


34 


27 


24954 


96837 


397132 


8 


n 


986039 


•54 


41 1092 


8.71 


588908 


33 


28 


24982 


96829 


397621 


8 


16 


986007 


•54 


4ii6i5 


8-70 


588385 


32 


29 


23010 


96822 


3 9 8iu 


8 


13 


985974 


•54 


412137 


8-69 


587864 


31 


30 
31 


25o38 
25o66 


9 68i5 


398600 


8 


14 


985942 


•54 
• 55 


412658 


8-68 


587342 


30 


96807 


9-399088 


8 


i3 


9 -985 9 o 9 


9-413179 


8-67 


io-58682i 


29 


32 


25og4 


96800 


399575 


8 


12 


986876 


• 55 


413699 


8-66 


5863oi 


28 


33 


2DI 22 


96793 


400062 


8 


11 


9 85843 


• 55 


414219 


8-65 


585 7 8i 


27 


34 


25i5i 


96786 


400549 


8 


10 


9 858u 


• 55 


414738 


8-64 


585262 


26 


35 


25179 


96778 


4oio35 


8 


3 


985778 


• 55 


4i5257 


8-64 


584743 


25 


36 


20207 


96771 


4oi52o 


8 


985745 


• 55 


4i5775 


8-63 


58,4225 


24 


37 


25235 


96764 


4o2oo5 


8 


07 


985712 


• 55 


416293 


8-62 


583707 


23 


3S 


25263 


96756 


402489 


8- 


06 


985679 


• 55 


416810 


8-6i 


583190 


22 


39 


22291 


96749 


402972 


8- 


o5 


983646 


• 55 


417326 


8-6o 


582674 


21 


40 
41 


25320 

75348 


96742 


4o3455 


8 


04 


980613 

9-985580 


• 55 

• 55 


417842 
9 -4i 8358 


8-5 9 

"8-58 


582i58 
10- 581642 


20 
19 


96734 


9-4o3938~ 


8- 


o3 


42 


25376 


96727 


404420 


8- 


02 


985547 


• 55 


418873 


8-57 


58 11 27 


18 


43 


25404 


96719 


404901 


8 


01 


9855i4 


• 55 


419387 


8-56 


58obi3 


17 


44 


25432 


96712 


405382 


8- 


00 


985480 


• 55 


4 1 990 1 


8-55 


580099 
579585 


16 


45 


2546o 


96705 


403862 


7- 


99 


985447 


• 55 


42041 5 


8-55 


15 


46 


25488 


96697 


4o634i 


7- 


98 


9 854i4 


• 56 


420927 


8-54 


579073 
5 7 856o 


14 


47 


255i6 


96690 


406820 


7- 


97 


9 853 80 


• 56 


421440 


8-53 


13 


48 


25545 


96682 


407299 


7- 


96 


985347 


• 56 


421932 


8-52 


578048 


12 


49 


25573 


96675 


407777 


7- 


9 5 


9853U 


• 56 


422463 


g-5i 


5 77 537 


11 


50 
51 


256oi 

2562g 


96667 


408254 


7" 


94 
94 


980280 

^985247" ' 


• 56 
^56 


422974 


8-5o 


577026 


10 
~9~ 


96660 


9-408731 


7- 


9 •423484 


~< ^7 


io-5765i6 


52 


25657 


96653 


409207 


7- 


93 


9852i3 


•56 


423993 


8-48 


576007 


8 


53 


25685 


96645 


409682 


7- 


92 


985i8o 


•56 


4245o3 


8-48 


575497 
574989 


7 


54 


25 7 i3 


9 6638 


410157 


7 


91 


985146 


•56 


425ou 


8-47 


6 


55 


25741 


96630 


4io632 


7 


00 


9 35 11 3 


•56 


4255i9 


8-46 


574481 


5 


56 


25769 

25798 


96623 


41 1 106 


7 


1 


985079 


• 56 


426027 


8-45 


573973 


4 


57 


966 1 5 


4ii579 


7 


985o45 


• 56 


426534 


8-44 


573466 


3 


58 


25826 


96608 


412052 


7 


87 


985oi 1 


•56 


427041 


8-43 


572909 


2 


59 


25854 


96600 


412524 


7 


86 


984978 


•56 


427547 


8-43 


572453 


1 


60 


25882 


96593 


412996 


7-83 


984944 


• 56 


428o52 


8-42 


571948 





N. cos. 


N. sine. 


L. cos. 


D.I'M 


L. sine. 




L. cot. | 


D. V 


L. tang. ' 








Tf&° 









SINES AND TANGENTS. — 15°. 



33 



Rad. = 1. 






Logarithms. — Radius = 


-10°. 




' 


N. sine. ! N. cos. 


L. sine. 


D.l" 


L. cos. 


D.l" 


L. tang. 


D.l" 


L. cot. 







25882 ; 96D93 


9-412996 


7 


-85 


9-984944 


'V 


9-428052 


8-42 


10-571948 j 60 


1 


25910 [96535 


413467 
4i3938 


7 


-84 


984910 


.57 


428557 


8-41 


571443 59 


2 


25938 96378 


7 


83 


984876 


•57 


429062 


8-4o 


570938 58 


3 


25966 96570 


414408 


7 


83 


984842 


•57 


429566 


8-3 9 
8-38 


570434 57 


4 


25994 i 96562 


414878 


7 


82 


984808 


•57 


430070 


569930 


56 


5 


26022 


9 6555 


4i5347 


7 


81 


984774 


•57 


43o573 


8-38 


569427 


55 


6 


26o5o 


96547 


4i58i5 


7 


80 


984740 


•57 


431070 


8-3 7 


568925 


54 


7 


26079 


96540 


416283 


7 


79 


984706 


•57 


43 1 577 


8-36 


568423 


53 


8 


26107 


96532 


416751 


7 


78 


984672 


•57 


432079 


8-35 


567921 


52 


9 


26i35 


96524 


417217 


7 


77 


984637 


.57 


43258o 


8-34 


567420 


51 


10 
11 


26i63 
26191 


96517 
96509 


417684 


7 


76 


984603 


•57 


433o8o 


8-33 


566920 
io-56642o 


50 
49 


9-4i8i5o 


7 


7 5 


9-984569 


.57 


9-43358o 


8-32 


12 


26219 


96502 


4i86i5 


7 


74 


984535 


•57 


434o8o 


8-32 


565920 


48 


13 


26247 


96494 


419079 


7 


73 


984500 


-07 


434579 


8-3i 


565421 


47 


14 


26275 


96486 


419544 


7 


73 


984466 


.57 


435078 


8-3o 


564922 


46 


15 


263o3 


96479 


420007 


7 


72 


984432 


•58 


435576 


8-29 


564424 


45 


16 


2633i 


96471 


420470 


7 


7i 


984397 


•58 


436073 


8-28 


563927 


44 


17 


26359 


96463 


420933 


7 


70 


984363 


-58 


436570 


8-28 


56343o 


43 


18 


26387 


96456 


421395 


7 


69 


984328 


-58 


437067 
437563 


8-27 


562933 


42 


19 


26415 


96448 


421857 
4223i8 


7 


68 


984294 


-58 


8-26 


562437 


41 


20 
21 


26443 


96440 


7 


67 


984209 


• 58 


438o59 


8-23 


561941 


40 


26471 j 96433 


9.422778 


7 


67 


9-984224 


• 58 


9^438554 


8-24 


io-56i446 


39 


22 


265oo 596423 


423238 


7 


66 


984190 


• 58 


439048 


8-23 


560952 


38 


23 


26528 96417 


423697 


7 


65 


984i55 


• 58 


439543 


8-23 


56o437 


37 


24 


26556 96410 


424i56 


7 


64 


984120 


• 58 


44oo36 


8-22 


559964 


36 


25 


26584 i 96402 


4246i5 


7 


63 


984085 


• 58 


440529 


8-21 


559471 


35 


26 


26612 (96394 


425073 


7 


62 


984050 


• 58 


441022 


8-20 


558978 


34 


27 j 


26640 9 6386 


42553o 


7 


61 


984015 


• 58 


44i5i4 


8.19 


558486 


33 


28 


26668 96379 


425987 


7 


60 


983981 


• 58 


442006 


8-19 


557994 


32 


29] 


26696 


96371 


426443 


7 


60 


983946 


• 58 


442497 


8-i8 


5575o3 


31 


sol 

31 


26724 


9 6363 


426899 


7 


5 9 


98391 1 


• 58 


442988 


8-i 7 


557012 


30 


26752 


96355 


9-427354 


7 


58 


9-983875 


• 58 


9-443479 


8-i6 


10-556521 


29 


32 


26780 


96347 


427809 


7 


57 


9 8384o 


• 5 9 


443968 


8-i6 


556o32 


28 


33 


26808 


96340 


428263 


7 


56 


9 838o5 


• 5 9 


444458 


8-i5 


555542 


27 


34 


26836 


96332 


428717 


7 


55 


983770 


• 5 9 


444947 


8-14 


555o53 


26 


35 


26864 


96324 


429170 


7 


54 


983735 


•5 9 


445435 


8-i3 


554565 


25 


36 


26892 


9 63 1 6 


429623 


7 


53 


983700 


• 5 9 


445923 


8-12 


554077 


24 


37 i 


26920 96308 


430075 


7 


52 


9 83664 


• 5 9 


44641 1 


8-12 


55358 9 


23 


ssj 


26948 96301 


43o527 


7 


52 


983629 


•5 9 


446898 


8-u 


553io2 


22 


39 j 


26976 96293 


430978 


7 


5i 


983594 


• 5 9 


447384 


8-io 


5526i6 


21 


40! 
411 


27004 


96285 


431429 


7 


5o 


9 83558 
9-983523 


•5 9 
-5 9 


447870 
9-448356 


8-09 


552i3o 


20 
19 


27032 


96277 


9-431879 


7 


49 


8-09 
8-o8 


io-55i644 


42 


27060 


96269 


432329 


7 


% 


983487 


• 5 9 


448841 


55i 169 


18 


43 


27088 96261 


432778 


7 


983452 


-5 9 


449326 


8-07 


530674 


17 


44 


27116 96253 


433226 


7 


47 


9 834i6 


• 5 9 


449810 


8-o6 


550190 


16 


45 


27144 96246 


433675 


7 


46 


98338i 


•5 9 


400294 


8- 06 


549706 


15 


46 


27172 96238 


434122 


7 


45 


983345 


• 5 9 


450777 


8-o5 


549223 14 1 


47 


27200 96230 


434569 


7 


44 


983309 
983273 


•5o 


451260 


8-04 


548740 [18 


48 


27228 j 96222 


435oi6 


7 


44 


• 60 


45i743 


8-o3 


548257 12 


49 


27256 96214 


435462 


7 


43 


9 83238 


• 6o 


452225 


8-02 


54777 5 11 


50 
51 


27284 
27312 


96206 


435908 


7 


42 


983202 


• 6o 

• 6o 


452706 

9^453187" 


8-02 


547294 


10 


96198 


9-436353 


7 


41 


9-983166 


8-oi 


io-546Si3 


~f 


52 


27340 I 96190 
27368S 96182 


436798 


7 


40 


983i3o 


• 6o 


453668 


8-oo 


546332 


8 


53 


437242 


7 


4o 


983094 


-6o 


454i 48 


7'99 


545852 


7 


54 


27396 96174 


437686 


7 


ll 


983o58 


• 60 


454628 


7-99 


545372 


6 


55 


274241 96166 


438129 


7 


983022 


• 6o 


455107 


7.98 


544893 


5 


56 


27452 96158 


4385 7 2 


7 


37 


982986 


• 6o 


455586 


7-97 


5444r4 


4 


57 


27480 96150 


439014 


7 


36 


982950 


• 6o 


436064 


7.90 


543g36 


3 


58 


27508 96142 


439456 


7 


36 


982914 


-6o 


456542 


7 - 9 6 


543458 


2 


59 


27536 96134 


439897 


7 


35 


982878 


• 6o 


457019 


7.93 


542981 


1 


60 


S 27564-96126 


43o338 


7 34 


982842 


• 6o 


437496 


7-94 


5425o4 
L. tang. ' 


N. cos. N. sine. 


L. cos. 


D.l" 


L. sine. 




L. cot. 


D.l" 


¥4° 



34: 



SINES AND TANGENTS. 



16 c 





Rad. = 1. 




Logarithms. — Radius = 


10 ] °. 




' 


N.sine. N. cos. 


L. sine. 


D. 1" 


L. cos. D.l" 


L. tang. 


D. 1" 


L. cot. 







27D64 196126 


9-44o338 


7-34 


9-982842 


• 6o 


9-457496 


7-94 


10- 5425o4 


60 


1 


27692 ; 96118 


440778 


7- 


33 


982805 


• 60 


457973 


7-g3 


542027 


59 


2 


27620 961 10 


441218 


7- 


32 


982769 
982733 


• 61 


458449 
458925 


7- 9 3 


54i55i 


58 


3 


27648 96102 


44i658 


7 


3i 


• 61 


7.92 


541075 


57 


4 


27676 96094 
27704 ! 96086 


442096 


7 


3i 


982696 


• 61 


459400 


7.91 


540600 


56 


5 


442535 


7 


3o 


982660 


• 61 


459875 


7'9° 


540125 


55 


6 


27731 96078 


442973 


7 


29 


982624 


• 6i 


460349 


7-90 


539651 


54 


7 


27709 96070 


4434io 


7 


28 


982587 


• 61 


460823 


7.89 


539177 


53 


8 


27787 96062 


443847 


7 


27 


982551 


• 61 


461297 


7-88 


538703 


52 




27813 96004 


444284 


7 


27 


982514 


.61 


461770 


7.88 


53823o 


51 


10 S 

11 


27843 
27871 


96046 


444720 


7 


26 


982477 


•61 


462242 


7-87 


537758 


50 


96037 


9-445i55 


7 


25 


9-982441 


"67 


9.462714 


7.86 


10-537286 


49 


12 


27899 


96029 


445590 


7 


24 


982404 


• 6i 


463 1 86 


7-85 


5368U 


4S 


18 1 


27927 


96021 


446025 


7 


23 


982367 


-6i 


463658 


7-85 


536342 


47 


14 


27955 ; 96013 


446459 


7 


23 


9 8233i 


-6i 


464129 


7-84 


535871 


46 


15 


2 79 83 


96005 


446893 


7 


22 


982294 


.61 


464599 


7-83 


5354oi 


45 


16: 


2801 1 


95997 


447326 


7 


21 


982237 


.61 


465069 


7-83 


53493i 


44 


17! 


28039 


95989 


447759 


7 


20 


982220 


.62 


465539 


7.82 


534461 


43 


IS' 


28067 


93981 


448191 


7 


20 


982183 


•62 


466008 


7.81 


533992 


42 


19 


28095 


95972 


448623 


7 


19 


982146 


.62 


466476 


7-80 


533524 


41 


20 


28123 


95964 


449054 


7 


18 


982109 


.62 


466945 


7-80 


533o55 


40 
39 


2l" 


28T57 


95956 


9.449483 


7 


17 


9-982072 


•62 


9-467413 


7-79 


io-532587 


22 


28178 


95948 


44991 5 


7 


16 


982035 


•62 


467880 


7.78 


532120 


38 


23 


28206 


93940 


45o345 


7 


16 


981998 


•62 


468347 


7-78 


53 1 653 


37 


24 


28234 


9 5 9 3i 


450775 


7 


i5 


981961 


•62 


468814 


7-77 


53 1 1 86 


36 


25 


28262 


95923 


45 i 204 


7 


14 


981924 


•62 


469280 


7-76 


53072b 


35 


26 


28290 


959 1 5 


45i632 


7 


i3 


9S1886 


.62 


469746 


7 - 7 5 


53o254 


34 


27 


283i8 


95907 


452o6o 


7 


i-3 


981849 


•02 


470211 


7.75 


529789 


33 


28 


28346 


93898 


452488 


7 


J2 


981812 


•62 


470676 


7-74 


529324 


32 


29 


28374 


93890 


452915 


7 


II 


981774 


•62 


47"4i 


7-73 


528859 


31 


j 30 

31 


28402 
28429 


9 5882 
958j4 


453342 
9-453768 


7 


10 


9 8i 7 3 7 


•62 


471605 
9-472068 


7.73 


528395 
10-527932 


30 
29 


7 


10 


9-981699 


•63 


7-72 


32 


28457 


95865 


454194 


7 


09 


981662 


•63 


472532 


7-71 


527468 


28 


3.3 


28485 


93857 


454619 


7 


08 


981625 


•63 


472993 


7.71 


527005 


27 


34 


285 1 3 


93849 


455o44 


7 


07 


981587 


•63 


473457 


7-70 


526543 


26 


35 


28541 


95841 


455469 


7 


07 


981549 


•63 


473919 


7-69 


526081 


25 


36 


2856 9 


95832 


455893 


7 


06 


98131 2 


•63 


474881 


7-69 


525619 


24 


37 


285 97 


95824 


4563 1 6 


7 


o5 


981474 


•63 


474842 


7-68 


525i58 


23 


38 


2->625 95816 


456739 


7 


04 


981436 


•63 


4753o3 


7-67 


524697 


22 


39 


28652 i 95807 


457162 


7 


04 


981399 


•63 


475763 


7-67 


524237 


21 


40 
41 


28680 ; 95799 


457584 


7 


o3 


98 1 36 1 


•63 


476223 


7-66 


523777_ 

io-5233i7 
522858 


20 
19~ 


28708 : 95791 


9^4580.06 


7 


02 


9-981323 


~63" 


9^476683 


7-65 




28736 95782 


458427 


7 


01 


981283 


•63 


477142 


7-65 


18 


43 


! 28764 


95774 


458848 


7 


01 


981247 


•63 


477601 


7-64 


522399 


17 


44 


; 28792 


95766 


459268 


7 


00 


981209 


• 63 


478059 


7-63 


52 1 941 


16 


1 45 


2 £8.20 


95757 


439688 


6 


99 


98-1171 


•63 


4785i 7 


7-63 


521483 


15 


4-; 


28847 93749 


460108 


6 


98 


9 8u33 


.64 


478973 


7-62 


521025 


14 


47 


1.28875 


95740 


460527 


6 


98 


98 1 095 


•64 


479432 


7-6. 


52o568 1 13 


4ft 


: 28903 


93732 


460946 


6 


97 


981057 


•64 


479889 


7-6i 


52oi 1 1 j 12 


49 


i 28931 


9 5 7 24 


461 364 


6 


96 


981019 


•64 


48o345 


7-60 


519655 


11 


N" 


j 28999 
1 28987 


9 J 7'3 
93707 


461782 
9-462199 


6 


93 


980981 


•64 
•64 


480801 
9 -481207,; 


7?fe -: 
7-5 9 


5 19 199 

io-5i8743 


10 j 
9 


6 


• 9 5 


9-980942 




! 290l5 


93698 


462616 


6 


94 


980904 


• 64 


481712 


7-58 


5i8 2 88 


8 


53 


29042 


95690 


463o32 


6 


•93 


980866 


.64 


482167 


7.57 


5i7833 


7 


54 


; 29O7O 


93681 


463448 


6 


•93 


9S0827 


•64 


482621 


7-57 


517379 


6 


55 


. 29O98 


9 56 7 3 


463864 


6 


.92 


980789 


•64 


483075 


7-56 


516925 


5 


56 


i 291 26 


9'>664 


464279 


6 


.91 


980750 


•64 


483529 


7 • 55 


516471 


4 


57 


29l54 


g5656 


464694 


6 


.90 


980712 


•64 


488982 


7-55 


5i6oi8 


3 


58 


29182 


93647 


465io8 


6 


X 


980673 


•64 


484435 


7 -54 


5 i 5565 


2 


59 


29209 


9563g 


465522 


6 


980635 


•64 


484887 


7-53 


5i5n3 


1 


60 


29237 


g563o 


j 465935 


6-88 


980396 


•64 


485339 


7-53 


5i466i 
L. tang. 







N. cos. N. sine. 


] L. cos. 


D. 1" 


L. sine. 




L. cot. 


D. 1" 






73° 







SINES AND TANGENTS. 17 c 



35 



Rad. = 1. 






Logarithms. — Radius = 


10 10 . 




, i 


N.sine.' N. cos. 


L. sine. 


D. 1" 


L. cos. 


D.l" 


L. tang. 


D. 1" 


L. cot. 




29237 t 9563o 


9-465935 


6-88 


9-980596 


•64! 


9-485339 


7-55 


10O14661 


60 


1 : 2g265 9D622 


466348 


6-88 


98o558 


•64 


485791 


7-52 


514209 
5i3 7 58 


59 


2 


29293 i 9061 3 


466761 


6-87 


980519 


•65 


486242 


7 -5i 


58 


8. 


29321 ! 9D605 


467173 


6-86 


980480 


•65 


486693 


7 -5i 


5i33o7 


57 


4' 


29348 J 93396 


467583 


6-85 


980442 


•65; 


487143 


700 


612837 


56 


5 


29376 


95588 


467996 


6-85 


980403 


•65' 


487593 


7-49 


512407 


55 


6 


29404 


95579 


468407 


6-84 


98o364 


•65; 


488043 


7-49 


011957 


54 


7 


29432 


95571 


468817 


6-83 


980326 


•65 


488492 


7-48 


5n5o8 


53 


8 


29460 


95562 


469227 


6-83 


980286 


• 65 


488941 
489390 


7-47 


5uo59 


52 


91 


29487 


95554 


469637 


6-82 


980247 


•65 


7-47 


5io6io 


51 


10 


2931 5 ! 93343 


470046 


6-8/1 


980208 


•65 


48 9 838 


7-46 


5ioi62 


50 


29543 i 93336 


9-470455 


6-8o 


9-980169 


• 65 


9-490286 


7-46 


10-509714 


49 


12 


29571 1 95328 


470863 


6-8o 


980130 


• 65; 


490733 


7-45 


509267 


48 


13 1 


29599 | 95519 


471271 


6-79 


980091 


•65 1 


491 180 


7-44 


308820 


47 


14 


29626 955i 1 


471679 


6-78 


980052 


•65 


491627 
492073 


7-44 


5o8373 


46 


15 


29654 \ 93502 


472086 


6-78 


980012 


•65 i 


7-43 


507927 


45 


16 


29682 95493 


472492 


6.77 


979973 


• 65 


492519 


7-43 


507481 


44 


17 


29710 1 93485 


472898 


6-76 


979934 


• 66 


492965 


7-^2 


5o7o35 


43 


18 


29737 93476 


4733o4 


6-76 


979895 


• 66 


493410 


7-4i 


506590 


42 


19 


29765 95467 


473710 


6. 7 5 


979855 


• 66 


493854 


7-40 


506146 


41 


20; 
"2T 


29793 1 95459 


474n5 


6-74 


979816 


• 66 


494299 


7.40 


5o57oi 


40 


29821 


9545o 


9-4745i9 


6-74 


9-979776 


•66! 


9-494743 


7-4o 


io-5o5257 


39 


22 


29849 


95441 


474923 
475327 


6-73 


979737 


•66! 


495i86 


7.39 


504814 


38 


23 


29876 


95433 


6-72 


979697 


• 66' 


49563o 


7-38 


504370 


37 


24 


29904 


95424 


475730 


6-72 


979658 


■66 | 


496073 


7.37 


503927 


36 


25, 


29932 


954i5 


476i33 


6.71 


9796-18 


• 66 


4965 1 5 


7-37 


5o3485 


35 


26 


29960 


95407 


476536 


6»7Q 


979579 


• 66 


496957 


7-36 


5o3o43 


34 


27; 


29987 


95398 


476938 


6-69 


979339 


• 66 j 


497399 


7-36 


502601 


33 


28' 


3ooi5 


95389 


477 3 4o 


6-69 


979499 


• 661 


497841 


7-35 


5o2i59 


32 


29 


3oo43 


9538o 


477741 


6-68 


979409 


-66 


498282 


7-34 


601718 


31 


30 

81 


30071 


95372 


478142 
9-478542 


6-67 


979420 
9-979380 


•66 | 

• 66 


498722 


7-34 


501278 
io-5oo837 


30 
29 


30098 95363 


6-67 


9-499163 


7-33 


32 


30126 J 95354 


478942 


6-66 


979340 


■66 


499603 


7-33 


500397 


28 


33 


3oi54 95345 


479^42 


6-65 


979300 


•67 


5ooo42 


7-32 


4999° 8 


27 


34 


3oi82 95337 


479741 


6-65 


979260 


•67 


500481 


7-3i 


499519 


26 


35 


30209 ] 95328 


480 1 40 


6-64 


979220 


•67 


500920 
5oi359 


7-3i 


499080 


25 


36 


30237 ; 95319 


480539 


6-63 


979180 


•67 


7>3o 


498641 


24 


37 


3o265 ! 95310 


480937 


6-63 


979140 


•67 


501797 


7>3o 


498203 


23 


38 


30292 953oi 


48i334 


6-62 


979100 


•67 


5o2235 


III 


497765 


22 


39 


3o32o l 95293 


48i 7 3i 


6-6i 


979059 


•67 


502672 


497328 


21 


40 


3o348| 93284 
30376 95275 


482128 
9-482525 


6-6i 
~6-6o 


979019 

9.978979 


•67 
•67 


5o3i09 


7-28 


496891 


20 


9-5o3546 


7.27 


10-496454 


19 


42 


3o4o3 ; 95266 


482921 


6-5o 


978939 


•67 


503982 


7-27 


4960 1 8 


IS 


48 


3o43 1 i 95257 


4833 16 


6-5 9 
6-58 


978898 


.67 


5o44i8 


7-26 


495582 


17 


44 


30459 95248 


483 7 i2 


978858 


•67 


5o4854 


7-25 


496146 


16 


45 


304861 95240 


484107 


6-5 7 


978817 


.67 


505289 


7-25 


4947 1 1 


15 


46 


3o5i4 95231 


4845oi 


6-57 


978777 


•67 


505724 


7-24 


494276 


14 


47 


3o342 ! 95222 


484895 


6-56 


978736 


•67 


5o6i59 


7.24 


493841 


13 


48 


3o57o 95213 


485289 


6-55 


978696 


•68 | 


506393 


7-23 


493407 


12 


49 


30597 95204 


485682 


6-55 


978655 


•68 


307027 


7-22 


492973 


11 


50 
~5T 


3o625 95195 
3o653 j 95186 


486075 


6-54 


978615 


•68 

•68 | 


007460 


7-22 


492540 


10 


9 • 486467 


6-53 


9-978574 


9007893 


~~7^r 


10-492107 


~T 


52 


3o68o: 95177 


486860 


6-53 


978533 


•68! 


5o8326 


7.21 


491674 


8 


53 


3o 7 o8 9 5i68 


487251 


6-52 


978493 


•68 


508759 


7 -20 


491241 


7 


54 


30736 95159 


487643 


6-5i. 


978452 


• 68 


509191 


7.19 


490809 
490378 


6 


55 


30763 931 5o 


488o34 


6-5i 


97841 1 


• 68 


509622 


7.19 

7-iS 


5 


56 


30791 93142 


488424 


6-5o 


97S370 


• 68 


5ioo54 


480946 


4 


1 *~ 


3o8i 9 95i33 


488814 


6-5o 


978329 


•68 


5 1 0485 


7.18 


489515 


8 


\ ^8 


30846 95124 


489204 


6-49 
6-48 


978288 


• 68 


510916 


7->7 


489084 


2 


59 


30874 S 931 i5 


489593 


978247 


• 68 


5n346 


7.16 


488634 


1 


60 


30902 
N. cos. 


95io6 


489982 


6-48 


978206 


• 68 


511776 


7.16 


488824 





N. sine. 


L. cos. 


D. 1" 


L. sine. 


L. cot. 


D.1" 


L. tang. 


72° 



36 



SINES AND TANGENTS. — 18°. 



Rad.=1. 






Logarithms. — Radius = 


10'°. 









N. sine. 


N. cos. 


L. sine. 


D. 1" 


L. cos. 


D.l" 


L. tang. 


D. 1" 


L. cot 




30902 


95ro6 


9-489982 


6-48 


9-978206 


-68 


9-611776 


7-16 


10-488224 


60 


1 


30929 


9 5o88 


490371 


6 


•48 


978165 


•68 


512206 


7.16 


487794 


59 


2 


30957 


490769 


6 


•47 


978124 


•68 


512635 


7.I.5 


487365 


58 


S 


30985 


95079 


491U7 


6 


-46 


978083 


.69 


5i3o64 


7-i4 


486o36 
486607 


57 


4 


3lOI2 


96070 


49i535 


6 


•46 


978042 


•69 


5i3493 


7-U 


56 


5 


3 1040 


95o6i 


491922 


6 


•45 


978001 


-6 9 


513921 
514349 


7-i3 


486079 


55 


6 


3 1 068 


95o52 


492308 


6 


•44 


977959 


.69 


7.1,3 


485651 


54 


7 


31095 


96043 


492605 
493081 


6 


44 


977918 


.69 


5i4777 


7.12 


485223 


53 


8 


3i 123 


95o33 


6 


■43 


977877 


-69 


5i5204 


7.12 


484796 


52 


9 


3ii5i 


95024 


493466 


6 


42 


977835 


.69 


5i563i 


7-n 


484369 
483 9 43 


51 


10 
11 


31178 


95oi5 


49385i 


6 


42 


977794 
9.977752 


.69 
.69 


5i6o57 


7- 10 


50 


3i2o6 


95oo6 


9>494236 


6 


4i 


9-616484 


7.10 


io-4835i6 


49 


12 


3i233 


94997 


494621 


6 


41 


977711 


.69 


616910 


7-09 


483090 


48 


13 


3i26i 


94988 


496005 


6 


40 


977669 


.69 


5i7335 


7-oo 
7-08 


482665 


47 


14 


31289 


94979 


4 9 5388 


6 


39 


977628 


.69 


517761 


482239 
48i8i5 


46 


15 


3i3i6 


94970 


495772 


6 


3 9 


977386 


.69 


5i8i85 


7-08 


45 


16 


3 1 344 


94961 


496154 


6 


38 


977544 


•70 


5i86io 


7-07 


481390 


44 


17 


3i372 


94952 


496537 


6 


? 7 


9775o3 


•70 


519034 


7-06 


480966 


43 


18 


3 1 399 


94943 


496919 
497301 


6 


ll 


977461 


•70 


619458 


7-06 


480642 


42 


19 


31427 


94933 


6 


977419 


• 70 


519882 


7»o5 


4801 18 


41 


20 
21 


3i454 


94924 
949i5 


497682 


6 


36 


977377 


.70 


52o3o5 


7-o5 


479695 


40 
39 


31482 


9-498064 


6 


35 


9-977335 


.70 


9.520728 


7-04 


lo-479 2 72 
478849 


22 


3i5io 


94006 
94897 


498444 


6 


34 


977293 


.70 


52ii5i 


7-o3 


38 


23 


3 1 537 


498825 


6 


34 


9772D1 


.70 


521573 


7-o3 


478427 


37 


24 


3 1 565 


94888 


499204 


6 


33 


977209 


.70 


521995 


7-o3 


478006 


36 


25 


3 1 593 


94878 


499584 


6 


32 


977167 


.70 


522417 
522838 


7-02 


477583 


35 


26 


31620 


94869 


499963 


6 


32 


977125 


.70 


7 -02 


477162 


34 


27 


31648 


94860 


5oo342 


6 


3i 


977083 


.70 


623259 


7-01 


476741 


33 


28 


3i6 7 5 


9485 1 


500721 


6 


3i 


977041 


.70 


52368o 


7-01 


476320 


32 


29 


3i7o3 


94842 


501099 


6 


3o 


976999 


.70 


524100 


7-00 


476900 


31 


30 
31 


3i73o 


94832 


501476 


6 


29 


976937 


.70 


524620 


6-99 


476480 


30 


3^58 


94823 


9-5oi854 


6 


29 


9-976914 


•7° 


9-524939 


6-99 


10-476061 


29 


32 


31786 


94814 


502231 


6 


28 


976872 


•7i 


525359 

525 77 8 


6-98 


474641 


28 


33 


3i8i3 


948o5 


502607 


6 


28 


976830 


•7i 


6-98 


474222 


27 


34 


31841 


94795 


502984 


6 


27 


976787 


•7i 


626197 


6-97 


4738o3 


26 


35 


3 1 868 


94786 


5o336o 


6 


26 


976745 


•7i 


5266i5 


6-97 


473385 


25 


36 


31896 


94777 
94768 


5o3735 


6- 


26 


976702 


•7i 


527033 


6-96 


472967 


24 


37 


31923 


5o4no 


6- 


25 


976660 


•7i 


527451 


6-96 


472649 


23 


38 


31961 


94758 


5o4485 


6 


25 


976617 


•7i 


527868 


6- 9 5 


472132 


22 


39 


31979 


94749 


5o486o 


6 


24 


976574 


•7i 


528285 


6-95 


471715 


21 


40 
41 


32006 
32o34 


9474o 


5o5234 


6 


23 


976532 


•7i 


528702 


6- 9 4 


471298 


20 


94730 


9-5o56o8 


6- 


23 


9-976489 


•71 


9-529119 
529635 


6- 9 3 


10-470881 


19 


42 


32061 


94721 


505981 


6- 


22 


976446 


•7i 


6- 9 3 


470465 


18 


43 


32089 


94712 


5o6354 


6- 


22 


976404 


•7i 


529960 


6- 9 3 


470060 


17 


44 


32Il6 


94702 


506727 


6- 


21 


976361 


.71 


53o366 


6-92 


469634 


16 


45 


32144 


94693 


507099 


6- 


20 


9 763i8 


.71 


530781 


6-91 


460219 


15 


46 


32171 


94684 


507471 


6- 


20 


976275 


• 71 


53i 196 


6-91 


468804 


14 


47 


32199 


94674 


507843 


6- 


19 


976232 


.72 


53i6ii 


6-90 


4683 89 


13 


48 


32227 


94665 


5o82i4 


6- 


ig. 


976189 


.72 


532025 


6-90 


467976 


12 


49 


32254 


94656 


5o8585 


6- 


18 


976146 


.72 


532439 
532853 


6-89 


467661 


11 


50 
51 


32282 


94646 


508966 


6- 


18 


976103 


.72 


6-89 


467147 


10 


32309 


9463 7 


9-509326 


6- 


17 


9-976060 


.72 


9-533266 


6-88 1 


10-466734 


9 


52 


3233 7 


94627 


509696 


6- 


16 


976017 


.72 


533679 


6-88 


466321 


8 


53 


32364 


94618 


5ioo65 


6- 


16 


975974 


.72 


534092 


5-87 


466908 


7 


54 


32392 


94609 


5io434 


6 


i5 


975930 


• 72 


5345o4 


6-87 


465496 


6 


55 


32419 


94599 


5io8o3 


6- 


i5 


975887 


.72 


534916 


6-86 


465o84 


5 


56 


32447 


94590 


5i 1 172 


6 


U 


975844 


.72 


535328 


9-86 


464672 


4 


57 


32474 


9458o 


5 1 1 54o 


6- 


i3 


976800 


.72 


535739 


6-85 


464261 


3 


58 


32502 


94571 


61 1907 


6- 


i3 


97 5 7 57 


.72 


536i5o 


6-85 


46385o 


2 


59 


32529 


9456i 


512275 


6 


12 


97D714 


.72 


53656i 


6-84 


46343o 


1 


60 


32557 


94552 


512642 


6-12 


975670 


• 72 


536972 


6-84 


463028 





N. cos. 


N. sine. 


L. cos. 


D. 1" 


L. sine. 




L. cot. | 


D. 1" 


L. tang. 


1 








71° 









SINES AND TANGENTS. — 19 c 



37 



Rad. = 1. 


Logarithms. — Radius =10 10 . 





N. sine. 


N. cos. 


L. sine. 


D.l" 


L. cos. ] 


3.1" 


L. tang. 
9.536972 


Dl." 


Xi. cot. 




3255 7 


94552 


9-512642 


6-12 


9-975670 


•73 


6-84 


io-463o28 


60 


1 


32584 


94542 


5 1 3009 


6- 11 


975627 


.73 


537382 


6-83 


462618 


59 


2 


32612 


94533 


5i3375 


6-n 


975583 


•73 


537791 


6-83 


462208 


58 


3 


32639 


94523 


5i374i 


6-io 


975539 


•73 


538201 


6-82 


461798 


57 


4 


32667 


94514 


514107 


6-09 


975496 
975402 


• 7 3 


5386i 1 


6-82 


461389 


56 


5 


32694 


945o4 


5i4472 


6-09 


.73 


539020 


6-8i 


460980 


55 


6 


32722 


94495 


5i4837 


6-08 


975408 


•73 


539429 


6-8i 


460571 


54 


7 


32749 


94485 


5l02O2 


6-08 


975365 


•73 


539837 


6-8o 


46oi63 


53 


8 


32777 


94476 


5i5566 


6»07 


975321 


•73 


540245 


6-8o 


459755 


52 


9 


32804 


94466 


5i593o 


6-07 


975233 


•73 


54o653 


6-79 


45 9 347 


51 


10 

11 


32832 
32809 


94457 


516294 


6-06 


• 7 3 


541061 


6-79 


458939 


50 
49 


94447 
9 4438 


9.516657 


6-o5 


9.975189 
975145 


•73 


9-54U68 


6-78 


10-458532 


12 


32887 


517020 


6-o5 


•73 


541875 


6.78 


458i25 


48 


13 


32914 


94428 


517382 


6-o4 


975101 


•73 


542281 


6.77 


457719 


47 


14 


32942 


94418 


517745 


6-04 


975057 


•73 


542688 


6.77 


457312 


46 


15 


32969 


94409 


518107 
518468 


6-o3 


975oi3 


• 7 3 


543094 


6-76 


406906 


45 


16 


32997 


94399 


6-02 


974969 


•74 


543499 


6.76 


4565oi 


44 


17 


33o24 


94390 


518829 


6-02 


974925 


•74 


543905 


6-75 


456095 


43 


18 


33o5i 


9438o 


519190 


6-oi 


974880 


•74 


5443io 


6-75 


455690 


42 


19 


33079 


94370 


5i95oi 


6-oi 


974836 


•74 


5447i5 


6-74 


455285 


41 


20 
21 


33io6 


9436i 


519911 


6-oo 


974792 


•74 
•74 


545 1 1 9 


6-74 


45488i 


40 


33i34 


9435i 


9.520271 


6-oo 


9-974748 


9-545524 


6- 7 3 


I0-454476 


39 


23 


33i6i 


94342 


52o63i 


5.99 


974703 


•74 


545928 


6.73 


454072 


38 


23 


83 1 89 


94332 


520990 
521349 


ill 


974659 


•74 


54633i 


6-72 


453669 


37 


24 


332i6 


94322 


974614 


•74 


546735 


6-72 


453265 


36 


25 


33244 


943 1 3 


521707 


5.98 


974570 


•74 


547i38 


6*71 


452862 


35 


26 


33271 


943 o3 


522066 


5-97 


974525 


•74 


547540 


6.71 


452460 


34 


27 


33298 


94293 


522424 


5-96 


97448i 


•74 


547943 


6-70 


452057 


33 


28 


33326 


94284 


522781 


5- 9 6 


974436 


•74 


548345 


6-70 


45i655 


32 


29 


33353 


94274 


523i38 


5-95 


974391 


•74 


548747 


6-69 


45i253 


31 


30 
31 


3338i 
334o8 


94264 


523495 


5-95 


974347 


•75 


549149 


6-69 


45o85i 


30 
29~ 


94254 


9-523852 


5-94 


9-974302 


•75 


9-5495oo 


6-68 


io-45o45o 


32 


33436 


94245 


524208 


5-94 


974207 


•75 


54995i 


6-68 


450049 


28 


33 


33463 


94235 


524564 


5.93 


974212 


•75 


55o352 


6-67 


449648 


27 


34 


33490 


94225 


524920 


5- 9 3 


974167 


• 7 5 


550752 


6-67 


449248 
448848 


26 


35 


335i8 


942 1 5 


525275 


5.92 


974122 


• 7 5 


55u52 


6-66 


25 


36 


33545 


94206 


52563o 


5.91 


974077 


•75 


55i552 


6-66 


448448 


24 


37 


33573 


94196 


525 9 84 


5.91 


974032 


.75 


55ig52 


6-65 


448048 


T3 


38 


336oo 


94186 


526339 
526693 


5.90 


973987 


•75 


55235i 


6-65 


447649 


22 


39 


33627 


94176 


5.90 


973942 


•75 


55275o 


6-65 


44725o 


21 


40 
41 


33655 


94167 


527046 


5.89 


973897 


•75 


553 1 49 


6-64 


44685 1 


20 


33682 


94i57 


9.527400 


5-89 


9-973852 


•7 5 


9-553548 


6-64 


io-446452 


19 


42 


33710 


94147 


527753 


5-88 


973807 


•75 


553946 


6-63 


' 446o54 


18 


43 


33 7 37 


94i37 


528io5 


5-88 


973761 


•75 


554344 


6-63 


445656 


17 


44 


33764 


94127 


528458 


5.87 


973716 


.76 


554741 


6-62 


445259 


16 


45 


33792 


941 18 


528810 


5.87 


973671 


.76 


555i39 


6-62 


444861 


15 


46 


33819 


94108 


529161 


5-86 


973625 


•76 


555536 


6-6i 


444464 


14 


47 


33846 


94098 


5295i3 


5-86 


97358o 


•76 


555 9 33 


6-6i 


444067 


13 


48 


338 7 4 


94088 


529864 


5-85 


973535 


.76 


556329 


6-6o 


443671 


12 


49 


33901 


94078 


53o2i5 


5-85 


973489 


•76 


556725 


6-6o 


443275 


11 


50 


33929 


94068 


53o565 


5-84 


973444 


•76 


557121 


6-5 9 


442879 


10 
9 


33956 


94o5t' 


9.530915 


5-84 


9.973398 


•76 


9.557517 


6-59 


10-442483 


52 


33 9 83 


94049 


53i265 


5-83 


973352 


.76 


557913 


6-59 


442087 


8 


53 


34oi 1 


94039 


53i6i4 


5.82 


973307 


•76 


5583o8 


6-58 


441692 


7 


54 


34o38 


94029 


53ig63 


5.82 


973261 


.76 


558702 


6-58 


441298 


6 


55 


34o65 


94019 


5323i2 


5-8i 


9732i5 


.76 


559097 


6-57 


440903 


5 


56 


34093 


94009 


53266i 


5-8i 


973169 


.76 


559491 
559885 


6-57 


440509 


4 


57 


34120 


93999 


533009 


5-8o 


973124 


•76 


6-56 


440 1 1 5 


3 


58 


34147 


93989 


533357 


5.8o 


973078 


.76 


560279 


6-56 


439721 2 


59 


34n5 


93979 


533704 


5-79 

5- 7 8 


973o32 


•77 


| 560673 


6-55 


43 9 32 7 | 1 


60 


34202 


93969 


534o52 


972986 


•77 


] 56io66 


6-55 


438934I 




N. cos. 


N. sine. 


L. cos. 


D.l" 


L. sine. 




L. cot. 


D.l" 


L. tang. ' 


70° 



38 



SINES AND TANGENTS. 20 c 



Rad. = 1. 


Logarithms. — Radius = 10 !O . 


' 


1 

2 
3 
4 
5 
6 
7 
8 
9 

10 
11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 
-LI 
i-1 
43 
44 
45 
46 
47 
48 
49 
50 


N. sine. 


N. cos. 


L. sine. 


D 


1" 


L. cos. ] 


D.l" 


L. tang. 


D 


1" 


L. cot. 




34202 

34229 

34257 
34284 
343 II 
34339 

34366 
34393 
34421 
34448 
34475 


93969 
93959 
93949 
93939 
9 3 9 2 9 
9 3 9 i 9 
93909 
93899 
9 388 9 
93879 
93869 


9.534o52 
534399 
534745 
535092 
535438 
535 7 83 
536129 
536474 
5368i8 
537i63 
537507 


5 
5 
5 
5 
5 
5 
5 
5 
5 
5 
5 


78 
77 
77 
77 

7 A 

75 
74 
74 
73 
73 


9.972986 
972940 
972894 
972848 
972802 
972755 
972709 
972663 
972617 
972570 
972524 


•77 
•77 
•77 
•77 
•77 
•77 
•77 
•77 
•77 
•77 
•77 


9-56io66 
561459 
56i85i 
562244 
562636 
563028 
563419 
5638i 1 
564202 
5645o2 
564983 


6 
6 
6 
6 
6 
6 
6 
6 
6 
6 
6 


55 
54 
54 
53 
53 
53 

52 
52 

5i 
5i 

5o 


10-438934 
438541 
438i49 
43 77 56 
437364 
436972 
43658i 
436189 
435798 
4354o8 
435017 


60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 


345o3 
3453o 
34557 
34584 
34612 
34639 
34666 
34694 
34721 
34748 


9 385 9 
93849 
9 3839 
93829 
93819 
93809 
93799 
93789 
93779 
93769 


9.537851 
538194 
538538 
53888o 
539223 
539565 
539907 
540249 
540590 
540931 


5 
5 
5 
5 
5 
5 
5 
5 
5 
5 


72 
72 
71 
71 
70 
70 

69 

68 
68 


9.972478 
97243i 
972385 
972338 
972291 
972245 
972198 
972i5i 
972105 
972058 


•78 
.78 
•78 
•78 
•78 
•78 
.78 
.78 


9-565373 
565763 
566 1 53 
566542 
566932 
567320 
567709 
568098 
568486 
568873 


6 
6 
6 
6 
6 
6 
6 
6 
6 
6 


5o 
49 
49 
49 
48 
48 
47 
47 
46 
46 


10-434627 
434237 
433847 
433458 
433o68 
43268o 
432291 
431902 
43i5i4 
43 1 1 27 


49 
48 
47 
46 
45 
44 
43 
42 
41 
40 


34775 
34«o3 
3483o 
3485 7 
34884 
349 1'-* 
34939 
34966 

34993 
35o2i 


93759 
93748 
93738 
93728 
93718 
93708 
93698 
9 3688 
9 36 77 
93667 


9-541272 
54i6i3 
541953 
542293 
542632 
542971 
5433io 
543649 
543987 
544325 


5 
5 
5 
5 
5 
5 
5 
5 
5 
5 


67 
66 
66 
65 
65 
64 
64 
63 
63 


9-972011 
971964 
971917 
971870 
971823 
971776 
971729 
971682 
97i635 
97i588 


' 7 2 

.78 

' l8 o 

.78 
.78 
.78 

•79 
•79 
•79 
•79 


9-569261 
569648 
57oo35 
570422 
570809 
571195 
5 7 i58i 

572352 
572738 


6 
6 
6 
6 
6 
6 
6 
6 
6 
6 


45 

45 

45 

44 

44 

43 

43 

42 

42 • 

42 


10-430739 
43o352 
429965 
429578 
429191 
428805 
428419 
428o33 
427648 
427262 

10-426877 
426493 
426108 
425724 
42534o 
424956 
424573 
424190 
423807 
.423424 


39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 
19 
18 
17 
16 
15 
14 
13 
12 
11 
10 

9 
8 
7 
6 
5 
4 
5 
2 
1 



35o48 
35o75 
35io2 

! 35i3o 
35i57 

i35i84 
352U 
35239 

; 35266 

1 35293 


93657 
93647 
93637 
93626 
93616 
93606 
93596 
9 3585 
93575 
9 3565 


9 • 544663 
545ooo 
545338 
545674 
54601 1 
546347 
546683 
547019 
547354 
547689 


5 
5 
5 
5 
5 
5 
5 
5 
5 
5 


62 
62 
61 
61 
60 
60 

58 


9-97i54o 
971493 
971446 
971398 
97i35i 
97i3o3 
971256 
971208 
97 1 161 
971 1 1 3 


•79 
•79 
•79 
•79 
•79 
•79 
•79 
•79 
•79 
•79 


9.573123 
573507 
573892 
574276 
574660 
575044 
575427 
575810 
576193 
576576 


6 
6 
6 
6 

6 
6 
6 
6 
6 
6 


4i 
41 
40 
4o 

39 
3 9 

39 
38 
38 
37 


! 35320 
i 35347 
135375 
35402 
I 35429 
! 35456 
1 35484 
355 1 1 
35538 
35565 


9 3555 
93544 
93534 
93524 
935i4 
935o3 
93493 
93483 
93472 
93462 


9-548024 
54835g 
548693 
049027 
549360 
549693 
55oo26 
55o359 
550692 
55io24 


5 
5 
5 
5 
5 
5 
5 
5 
5 
5 


57 
5 7 
56 
56 
55 
55 
54 
54 
53 
53 


9-971066 
971018 

970970 
970922 

97° 8 74 
970827 

970779 
970731 
970683 
970635 


-8o 
-8o 
•80 
•80 
-8o 
•80 
•80 
•80 
•80 
-80 


9-576958 
577341 
577723 
578104 
578486 
578867 
579248 
579629 
580009 
58o389 


6 
6 
6 
6 
6 
6 
6 
6 
6 
6 


37 
36 
36 
36 
35 
35 
34 
34 
34 
33 


io-423o4i 

422609 
422277 
421896 
42i5i4 
42 1 1 33 
420752 
420371 

4I999 1 
419611 


51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


1 35592 
356 1 9 

I 35647 
35674 
35701 
35728 
35 7 55 

j 35782 
3 0810 
3583 7 


93452 
93441 
9343i 
93420 
93410 
93400 
93389 
g3379 
9 3368 
9 3858 


9 -55i356 
55i68 7 
552oi8 
552349 
55268o 
553oio 
55334i 
553670 
554ooo 
554329 


5 
5 

5 
5 
5 
5 
5 
5 
5 
5 


52 
52 
52 

5i 
5i 
5o 
-5o 

49 
49 
48 


9-970586 
970538 
970490 
970442 
970394 
970345 
970297 
970249 
970200 
970152 


•8o 

•80 
-8o 
-8o 
•80 
•81 
-8i 
-8i 
•81 
•81 


9-580769 
58 1 1 49 
58i528 
581907 
582286 
582665 
583o43 
583422 
5838oo 
584177 


6 
6 
6 
6 
6 
6 
6 
6 
6 
6 


33 

32 
32 
32 

3i 
3i 

3o 
3o 

29 
29 


io-4i923i 

4i885i 
418472 
418093 
4I77U 
417335 
416957 
416578 
416200 
4i5823 




1 N. cos. 


N. sine. 


L. cos. 


D.l" 


L. sine. 




L. cot. 


D.l" 


L. tang. 


' 


69° 



SINES AND TANGENTS. — 21°. 



39 




40 



SINES AND TANGENTS. — 22°. 



Rad. = 1. 






Logarithms.— 


-Radius = 10 10 . 




' N.sine. 


N. cos. 


L. sine. 


D. 1" 


L. cos. b.l" 

1 


L. tang. 


D. 1" 


L. cot. 







37461 


92718 


9-573575 


5 


21 


9-967166 


-85 


9-606410 


6-06 


10-393590 


60 


1 


37488 


92707 


073888 


5 


20 


9671 1 5 


-85 


606773 


6-06 


393227 


59 


2 


37Di5 


92097 


574200 


5 


20 


967064 


-85 


607137 


6«oo 


392863 


58 


3 


37342 


92686 


5745i2 


5 


19 


967013 


-85 


607600 


6-o5 


392500 


57 


4 


37069 


92670 


574824 


5 


19 


966961 


• 85 


607863 


6-04 


392137 


56 


5 


37390 


92664 


075i36 


5 


*9 


966910 


•85 


608226 


6-o4 


391775 


55 


6 


37622 


92653 


575447 


5 


18 


966809 


• 85 


608688 


6-o4 


391412 


54 


7 


37649 


92642 


575758 


5 


18 


966808 


-85 


608960 


6-o3 


391060 


53 


8 


37676 


9263i 


576069 


5 


17 


966756 


• 86 


609312 


6-o3 


390688 


52 


9 


37703 


92620 


676379 


5 


17 


. 966705 


• 86 


609674 


6-o3 


390326 


51 


10 
11 


3 77 3o 

37707 


926o 9 _ 


676689 


5 


16 


9 66653 


•86 


6ioo36 


6-02 


389964 


50 


92098 


9.576999 


5 


16 


9-966602 


•86 


9'6io397 


6-02 


10-389603 


4(5 


12 


37784 


92087 


577309 


5 


16 


96655o 


• 86 


610769 


6-02 


389241 


48 


13 


3 7 8u 


92076 


577618 


5 


i5 


966499 


• 86 


611120 


6-oi 


38888o 


47 


14 


37808 


92560 


577927 


5 


i5 


966447 


• 86 


61 1480 


6-oi 


388520 


46 


15 


37863 


92554 


578236 


5 


14 


966395 


• 86 


611841 


6-oi 


388i5 9 


45 


16 


37892 


92043 


578645 


5 


14 


966344 


• 86 


612201 


6-oo 


387799 


44 


17 


37919 


92532 


578853 


5 


i3 


966292 


• 86 


612661 


6-oo 


387439 


43 


18 


37946 


92021 


079162 


5 


i3 


966240 


• 86 


612921 


6-oo 


387079 


42 


19 


37973 


92610 


579470 


5 


i3 


966188 


• 86 


6i328i 


5-99 


386719 


41 


20 
21 


38026 


92499 


579777 


5 


12 


966136 


• 86 


6i364i 


5-99 


38635 9 


40 


92488 


9-58oo85 


5 


12 


9-966085 


.87 


9-614000 


5-98 


io-386ooo 


39 


22 


38o53 


92477 


58o3 9 2 


5 


11 


966033 


.87 


614369 


5.98 


385641 


38 


23 


38o8o 


92466 


580699 


5 


11 


960981 


.87 


614718 


5- 9 8 


385282 


37 


24 


38107 


92455 


58ioo5 


5 


11 


965928 


•87 


616077 


5.97 


384923 


36 


25 


38i34 


92444 


58i3i2 


5 


10 


963876 


.87 


61 5435 


5.97 


384565 


85 


26 


38i6i 


92432 


58i6i8 


5 


10 


965824 


.87 


616793 


0-97 


384207 


84 


27 


38i88 


92421 


581924 


5 


09 


965772 


•87 


6i6i5i 


5.96 


383849 


33 


2S 


382i5 


92410. 


582229 


5 


09 


966720 


.87 


616609 


6-96 


383491 


32 


29 


38241 


92399 


582535 


5 


09 


963668 


.87 


616867 


5-96 


383i33 


31 


30 
31 


38268 
38295 


92388 


582840 


5 


08 


963616 


.87 


617224 


0-90 


382776 


30 


92377 


9-583U5 


5 


08 


9-966363 


.87 


9-617682 


5-95 


io-3824i8 


29 


32 


38322 


92366 


583449 


5 


07 


966611 


.87 


617939 


5- 9 5 


382061 


28 


33 


38349 


92355 


583754 


5 


07 


963.568 


• 8i 


618296 


5-94 


381705 


27 


84 


38376 


92343 


584o58 


5 


06 


966406 


•87 


618662 


0-94 


38i348 


26 


35 


384o3 


92332 


58436i 


5 


06 


965353 


•88 


619008 


5-94 


380992 


25 


36 


3843o 


92321 


584665 


5 


06 


9653oi • 


• 88 


619364 


5- 9 3 


38o636 


24 


37 


38456 


92310 


584968 


5 


o5 


966248 


• 88 


619721 


5- 9 3 


380279 


23 


38 


38483 


92299 


585272 


5 


o5 


966196 


• 88 


620076 


5«93 


379924 


22 


39 


385io 


92287 


585574 


5 


04 


96614.3 


• 88 


620432 


5-92 


37 9 568 


21 


40 
41 


38537 
"38564" 


92276 
92265 


585877 


5 


04 


965090 


• 88 


620787 


6-92 


379213 


20 


9-586179 


5 


o3 


9-966037 


• 88 


9-621142 


5-92 


10-378868 


19 


42 


38591 


92254 


586482 


5 


o3 


964984 


-88 


621497 


6-91 


3785o3 


18 


43 


38617 


92243 


586783 


5 


o3 


964931 


-88 


621862 


5-91 


378148 


17 


44 


38644 


92231 


587085 


5 


02 


964879 


• 88 


622207 


5*90 


377793 


16 


45 


38671 


92220 


587386 


5 


02 


964826 


• 88 


622561 


6-90 


377439 


15 


46 


386 9 8 


92209 


58 7 688 


5 


01 


964773 


• 88 


622916 


5-oo 


377086 


14 


47 


38725 


92198 


587989 


5 


01 


964719 


•88 


623269 


5.89 


376731 


13 


48 


38 7 52 


92186 


588289 


5 


01 


964666 


.89 


623623 


5-8 9 


376377 


12 


49 


38778 


92175 


5885 9 o 


5 


00 


964613 


.89 


623976 
62433o 


5-8 9 


376024 


11 


50 
51 


388o5 
"3883T 


92164 


588890 


5 


00 


964660 


.89 
.89 


5-88 


375670 


10 


92l52 


9-589190 


4 


99 


9-964507 


9.624683 


5-88 


10-376317 


9 


52 


3885 9 


92141 


689489 


4 


99 


964454 


.89 


. 625o36 


5-88 


374964 


8 


53 


38886 


92i3o 


589789 


4 


99 


964400 


.89 


625388 


5.87 


374612 


7 


54 


38912 


92119 


590088 


4 


98 


964347 


.89 


626741 


5.87 


374259 


6 


55 


38939 


92107 


590387 


4 


98 


964294 


.89 


626093 


5.87 


ItycVr 


5 


56 


38966 


92096 


590686 


4 


97 


964240 


.89 


626445 


5-86 


373555 


4 


57 


38 99 3 


92085 


590984 


4 


97 


964187 


.89 


626797 


5-86 


373203 


8 


58 


39020 


92073 


591282 


4 


97 


964i33 


.89 


627149 


5-86 


372851 


2 


59 


39046 


92062 


591580 


4 


96 


964080 


.89 


627601 


5-85 


372499 
372148 


1 


60 


39073 
N. cos. 


92o5o 


591878 


4 


96 


964026 


.89 


627862' 


5-85 





N. sine. 


L. cos. 


D 


1" 


L. sine. 




L. cot. 


D. V 


L. tang. 


' 


«? 



SINES AND TANGENTS. 23°. 



41 



Rad. = 1. 


Logarithms. — Radius = lO 10 . 


/ 


N. sine. 


N. cos. 


L. sine. 


D. 1" 


L. cos. ] 


D.l" 


L. tang. 


D 


1" 


L. cot. 






1 

2 
8 
4 
5 
6 
7 
8 
9 
10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
85 
36 
37 
38 
39 
40 


39073 
39100 
39127 
39i53 
39180 
39207 
39234 
39260 
39287 
3g3i4 
39341 


92o5o 
92039 
92028 
92016 
92005 
91904 
91982 
91971 
9 i 9 5o 
91948 
91936 


9-591878 
592176 
592473 
592770 
593067 
593363. 
59365# 
593955 
594251 
594547 
594842 


4.96 
4-95 
4-95 
4-9 5 
4-94 
4.94 

4-93 
4- 9 3 
4-93 
4-92 
4-92 


9-964026 
963972 
963919 
963865 
963811 
963757 
963704 
96365o 
963596 
963542 
96a488 


.89 
.89 
.89 
-90 
•90 
.90 
.90 
•90 
•90 
•90 
.90 


9-627852 
628203 
628554 
628905 
629255 
629606 
629956 
63o3o6 
63o656 
63ioo5 
63i355 


5 
5 
5 

5 
5 
5 
5 
5 
5 
5 
5 


85 
85 
85 
84 
84 
83 
83 
83 
83 
82 
82 


10-372148 

371797 
371446 
371095 
370745 
370394 
370044 
369694 
369344 
368 99 5 
368645 


60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 


39367 
39394 
39421 
3 9 448 
39474 
39501 
39528 
39555 
39581 
39608 


91925 
919U 
91902 
91891 
91879 
91868 
9i856 
91845 
9i833 
91822 


9-595137 
595432 
595727 
596021 
5963i5 
596609 
696900 
597196 
597490 
5 9 7 7 83 


4- 9 i 
4-91 

4- 9 i 
4.90 
4.90 
4-89 
4-89 
4-8 9 
4-88 
4-88 


9-963434 
963379 
963325 
963271 
963217 
963i63 
963108 
963o54 
962999 
962945 


.90 
.90 
•90 
.90 
.90 
.90 
•91 
•91 
.91 
•91 


9-631704 
632o53 
632401 
632750 
633o 9 8 
633447 
633795 
634U3 
634490 
634838 


5 
5 
5 
5 
5 
5 
5 
5 
5 
5 


82 
81 
81 
81 
80 
80 
80 
79 
79 
79 


10-368296 
367947 
367599 
367250 
366902 
366553 
3662o5 
365857 
3655io 
365i62 


49 

48 
47 
46 
45 
44 
43 
42 
41 
40 


39635 
39661 
3 9 688 
39715 
3 97 4i 
39768 
3 979 5 
39822 
39848 
3 9 8 7 5 


91810 
91799 
91787 
91775 
91764 
91702 
9*741 
91729 
91718 
91706 


9-598073 
598368 
5 9 866o 
598952 
599244 
599536 
599827 
6001 18 
600409 
600700 


4-8 7 
4-8 7 
4-8 7 
4-86 
4-86 
4-85 
4-85 
4-85 
4.84 
4-84 


9-962890 
962836 
962781 
962727 
962672 
962617 
962562 
962508 
962453 
962398 


.91 
•91 

•91 
.91 

•91 
•91 
•91 
•91 
.91 
.92 


9 -635i85 
635532 
635879 
636226 
636572 
636919 
637265 
63761 1 
63 79 56 
638302 


5 
5 
5 
5 
5 
5 
5 
5 
5 
5 


78 
78 
78 
77 
77 
77 
77 

76 


10 -3648i 5 
364468 
364121 
363 77 4 
363428 
363o8i 
362 7 35 
36238 9 
362044 
361698 


39 
38 
37 
36 
35 
34 
33 
32 
31 
30 
29 
28 
27 
26 
25 
24 
23 
22 
21 
20 


39902 
39928 
39955 
39982 
40008 
4oo35 
40062 
40088 
4ou5 
40141 


91694 
9 i683 
91671 
91660 
91648 
9i636 
91625 
91613 
91601 
91590 


9-600990 
601280 
601570 
601860 
6o2i5o 
602439 
602728 
6o3oi8 
6o33o5 
603594 


4.84 
4-83 
4-83 
4-82 
4-82 
4-82 
4-8i 
4-8i 
4-8i 
4-8o 


9-962343 
962288 
962233 
962178 
962123 
962067 
962012 
961957 
961902 
961846 


.92 
•92 
•92 
•92 
•92 
.92 
•92 
.92 
.92 
.92 


9-638647 
638992 
639337 
63 9 682 
640027 
640371 
640716 
641060 
641404 
641747 


5 
5 
5 
5 
5 
5 
5 
5 
5 
5 


75 
■75 
75 
74 
74 
74 
73 
73 
73 
72 


io-36i353 
36ioo8 
36o663 
36o3i8 
359973 
359629 
359284 
358940 
3585 9 6 
358253 


41 
42 
43 
44 
45 
46 
47 
48 
49 
50 


40168 
40195 
40221 
40248 
40275 
4o3oi 
4o328 
4o355 
4o38i 
40408 


9 i5 7 8 
9 1 566 
9i555 
9i543 
9 1 53 1 
91519 
9i5o8 
91496 
91484 
9U72 


9-6o3882 
604170 
604457 
604745 
6o5o32 
6o53i9 
60D606 
605892 
606179 
606465 


4-8o 
4-79 
4-79 
4-79 
4-7» 
4-78 
4-78 
4-77 
4-77 
4-76 


9-96i79i 
961735 
961680 
961624 
961569 
961613 
961459 
961402 
961346 
961290 


•92 
.92 
.92 
. 9 3 
•93 
•93 
•93 
•93 
-98 

•93 


9-642091 
642434 
642777 
643120 
643463 
6438o6 
644148 
644490 
644832 
645174 


5 
5 
5 
5 
5 
5 
5 
5 
5 
5 


72 

72 
72 
71 
71 
71 
70 
70 
70 
69 


10-357909 
357566 
357223 
35688o 
356537 
356194 
355852 
3555io 
355i68 
354826 


19 
18 
17 
16 
15 
14 
13 
12 
11 
10 


51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


40434 
40461 
40488 
4o5l4 
4o54i 
40567 
40594 
4062I 
40647 
40674 


91461 
91449 
91437 
9U25 
9UU 
91402 
91390 
91378 
9i366 
9i355 


9-606751 
607036 
607322 
607607 
607892 
608177 
608461 
608745 
609029 
609313 


4-76 
4-76 
4-75 
4-75 
4-74 
4-74 
4-74 
4-73 
4-73 
4-73 


9-961235 
961 179 
961 123 
961067 
961011 
960955 
960899 
960843 
960786 
960730 


•93 
•93 
•93 
■93 
- 9 3 

•93 
• 9 3 

•94 
•94 
•94 


9-6455i6 
645857 
646199 
646540 
646881 
647222 
647562 
647903 
648243 
648583 


5 
5 
5 
5 
5 
5 
5 
5 
5 
5 


i 9 

i 9 
69 

68 
68 
68 

•67 
.67 
.67 
• 66 


io-354484 
354143 
3538oi 
35346o 
353ii9 
352778 
352438 
352097 
35i 7 5 7 
3514.17 


9 
8 
7 
6 
5 
4 
3 
2 
1 



N. cos. N. sine. 


L. cos. j D. 1" 


L. sine. 




j L. cot. 


D. 1" 


L. tang. 


; 


66° 



SINES AND TANGENTS. — 24°. 



Rad. = 1. 


Logarithms. — Radius = IO 10 . 


' 


N. sine. JN. cos. 


L. sine. 


D.l" 


L. cos. b.l"l 


L. tang. 


Dl." 


L. cot. 






1 

2 
3 
4 
5 
6 
7 
8 
9 
10 


40674 
40700 
40727 
40753 
40780 
40806 
4o833 
40860 
40886 
40913 
40939 


9 i355 
9i343 
9i33i 
91319 
91307 
91296 
91283 
91272 
91260 
91248 
91236 


9-609313 
609597 
609880 
610164 
610447 
610729 
611012 
611294 
611576 
6n858 
612140 


4- 7 3 
4-72 
4-72 
4-72 
4-71 
4-7i 
4-70 
4-70 
4-70 
4.69 
4.69 


9-960730 
960674 
960618 
960561 
96o5o5 
960448 
960392 
96o335 
960279 
960222 
960165 


.94 
•94 
.94 
.94 
•94 
.94 
.94 
•94 
.94 
•94 
.94 


9-648583 
648923 
649263 
649602 
649942 
65o28i 
65o62o 
650959 
651297 
65i636 
65 1 974 


5-66 
5-66 
5-66 
5-66 
5-65 
5-65 
5-65 
5-64 
5-64 
5-64 
5-63 


io-35i4i7 

351077 
350737 
35o398 
35oo58 

349719 
34938o 
349041 
348703 
348364 
348026 


60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 


11 

12 
13 
14 
15 
16 
17 
18 
19 
20 
21 
22 
23 
24 
25 
26 
27 
28 
29 
30 


40966 
40992 
41019 
4io45 
41072 
41098 
4ii25 
4ii5i 
41178 
41204 
4i23i 
41257 
41284 
4i3io 
4i337 
4i363 
41390 
41416 
41443 
41469 


91224 
91212 
91200 
91188 
91176 
91164 
91152 
91140 
91128 
91116 


9-612421 
612702 
612983 
613264 
6i3545 
6i3825 
6i4io5 
6U385 
6i4665 
6i4944 


4.69 
4-68 
4-68 
4-67 
4-67 
4-67 
4-66 
4-66 
4-66 
4-65 


9-960109 
960052 

$$ 

959882 
95982.5 
959768 
95971 1 
959654 
959D96 


- 9 5 
•9 

' 9 l 

• 9 5 

- 9 5 


9-6523i2 
65265o 
652 9 88 
653326 
653663 
654ooo 
654337 
654674 
655ou 
655348 


5-63 
5-63 
5-63 
5-62 
5-62 
5-62 
5-6i 
5-6i 
5- 61 
5-6i 


10-347688 
34735o 
347012 
346674 
346337 
346000 
345663 
345326 

344989 
344652 


49 
48 
47 
46 
45 
44 
43 
42 
41 
40 


91 104 
91092 
91080 
91068 
9io56 
91044 
91032 
91020 
91008 
90996 


9-6i5223 
6i55o2 
6i5 7 8i 
616060 
6i6338 
616616 
616894 
617172 
617450 
617727 


4-65 
4-65 
4-64 
4-64 
4-64 
4-63 
4-63 
4-62 
4-62 
4-62 


9-959539 
9D9482 
959425 
959368 
959310 
959253 
959195 
959138 
959081 
959023 


- 9 5 

' 9 l 

' 9 \ 

' 9 l 
.96 

.96 

.96 

.96 

.96 

.96 


9-655684 
656o2o 
656356 
656692 
657028 
657364 
657699 
658o34 
65836 9 
658704 


5-6o 
5-6o 
5-6o 
5-59 
5-5 9 
5-5 9 
5-5 9 
5-58 
5-58 
5-58 


io-3443i6 
343980 
343644 
3433o8 
342972 
342636 
3423oi 
341966 
34i63r 
341296 


39 
38 
37 
36 
35 
34 
33 
32 
31 
30 


31 
32 
33 
34 
35 
36 
37 
38 
39 
40 


41496 

4l522 

41 549 
41575 
41602 
41628 

4i655 
41681 
41707 
41734 


90984 
90972 
90960 
90948 
90986 
90924 
9091 1 
90899 
90887 
90875 


9-618004 
618281 
6 i 8558 
6i8834 
619110 
619386 
619662 
619938 
620213 
620488 


4-6i 
4-6i 
4-6i 
4-6o 
4-6o 
4-6o 
4.59 
4-59 
4-5 9 
4-58 


9-958965 
958908 
9 5885o 
958792 
958734 
958677 
958619 
9 5856i 
9 585o3 
958445 


.96 
•96 
.96 
.96 
.96 
.96 
.96 
.96 
•97 
•97 


9-659039 
65 9 373 
659708 
660042 
660376 
660710 
661043 
661377 
661710 
662043 


5-58 
5-5 7 
5-5 7 
507 
5-5 7 
5-56 
5-56 
5-56 
5-55 
5-55 


10-340961 
340627 
340292 
339958 
339624 
339290 
338 9 57 
338623 
338290 
337937 


29 
28 
27 
26 
25 
24 
23 
22 
21 
20 

T9| 
18 

16 
15 
14 
13 
12 
11 
10 


41 
42 
43 
44 
45 
46 
47 
48 
49 
50 


41760 

41787 
4i8i3 
41840 
41866 
41892 
41919 
41945 
41972 
41998 


9 o863 
9085 1 
90839 
90826 
90814 
90802 
90790 
90778 
90766 
90753 


9-620763 
62io38 
62i3i3 
621587 
621861 
622i35 
622409 
622682 
622956 
623229 


4-58 
4-57 
4-5 7 
4-57 
4-56 
4-56 
4-56 
4-55 
4-55 
4-55 


9 - 9 58387 
958329 
958271 
9582i3 
958i54 
958096 
9 58o38 
957979 
937921 
95 7 «63 


•97 
•97 
•97 
•97 
•97 
•97 
•97 
•97 
•97 
•97 


9-662376 
662709 
663o42 
663375 
663707 
664039 
664371 
664703 
665o35 
665366 


5-55 
5-54 
5-54 
5-54 
5-54 
5-53 
5-53 
5-53 
5-53 
5-52 


10-337624 
337291 
336 9 58 
336625 
336293 
335961 
335629 
335297 
334965 
334634 


51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


42024 
42o5i 
42077 
42104 
42i3o 
42 1 56 
42183 
42209 
42235 
42262 


90741 
90729 
90717 
90704 
90692 
90680 
90668 
90655 
90643 
9063 1 


9-6235o2 
623774 
624047 
624319 
624591 
624863 
625i35 
625406 
625677 
625948 


4-54 
4-54 
4-54 
4-53 
4-53 
4-53 
4-52 
4-52 
4-52 
4-5i 


9-957804 
907746 
957687 
957628 
957570 
95751 1 
957452 
957393 
957335 
957276 


' 9 l 
.98 

.98 

.98 

.98 

.98 

.98 

.98 

.98 

.98 


9-665697 
666029 
66636o 
666691 
667021 
667352 
667682 
6680 1 3 
668343 
668672 


5-52 
5-52 
5-5i 
5-5i 
5-5i 
5-5i 
5-5o 
5-5o 
5-5o 
5-5o 


io-3343o3 
333971 
333640 
333309 
332979 
332648 
3323i8 
331987 
33i657 
33i328 


9 
8 
7 
6 
5 
4 
3 
2 
1 



N. cos 


N.sine 


L. cos. 


D.l" 


L. sine. 




L. cot. 


D.l" 


L. tang. 


' 


<S5° , 



SINES AND TANGENTS. 25°. 



43 





Rad. = 


1. 


| 




Logarithms.— 


-Radius = 10'°. 




G 


N. sine. 


N. cos. 


] L. sine. 


D. 1" 


L. cos. 


D.l" 


L. tang. 


D 1" 


L. cot. 




42262 |9<)63? 


9-626948 


4-5i 


9.907276 


.98 


9-668673 


5-5o 


I0'33i327 


60 


I 


42288 90618 


626219 


4-5i 


907217 


.98 


669002 


5-49 


330998 


59 


2 


423 1 5 00606 


626490 


4-5i 


967168 


.98 


669332 


5-49 


33o668 


58 


3 


4234? 


90694 


626760 


4-5o 


967099 


.98 


669661 


5.49 


33o339 


57 


4 


4-2367 


90682 


627030 


4-5o 


967040 


.98 


669991 


5-48 


330009 


56 


o 


42394 


90669 


627800 


4-5o 


965981 


.98 


670320 


5.48 


829680 


55 


6 


42420 


90557 


627070 


4-49 


966921 


•99 


670649 


5.48 


329351 


54 


7 


42446 


00545 


627840 


4.49 


966862 


•99 


670977 


5.48 


329023 


53 


8 


42473 90082 


628109 


4-49 


906803 


•99 


671806 


5-47 


328694 


52 


S 


42499 i 90620 


628378 


4-48 


056744 


•99 


671634 


5-47 


328366 


51 


It) ' 

n 


4202 j ' 90007 


62^)647 


4-48 


966684 


•99 


671968 


5-47 


328037 


50 


42D32 90496 


9-628916 


4-47 


9-956620 


•99 


9.672291 


5-47 


10-327709 


4y 


12 


42578(00483 


629186 


4-47 


9 56566 


•99 


672619 


5-46 


32 7 38i 


48 


u 


42604 I 90470 


629453 


4.47 


966606 


'99 


672947 


5.46 


327003 


47 


14 


4263i 90408 


629721 


4.46 


956447 


'99 


673274 


5-46 


326726 


46 


15 


42057 ! 90446 


629989 


4.46 


966387 


'99 


673602 


5-46 


326398 


45 


16 ! 


42683 1 90433 


630267 


4.46 


966327 


•99 


673929 


5-45 


326071 


44 


17 


42709 90421 


63o524 


4.46 


966268 


'99 


674257 


5-45 


326743 


43 


18 


42736 1 90408 


630792 


4-45 


966208 


1 -00 


674584 


5-45 


326416 


42 


IS 


42762 00396 


631069 


4-45 


966148 


I -00 


674910 


5-44 


326090 


41 


20 
21 


42788 i 9 o383 


63i326 


4-45 


966089 


I -00 
I -00 


676237 


5-44 


324763 


40 
39 


42816 90371 


9 -631503 


4-44 


9-966029 


9-675664 


5-44 


10-324436 


22 


42841 903 58 


63 1 809 


4.44 


966969 


I -00 


676890 


5-44 


324110 


38 


23 


42867 j 90346 


632125 


4-44 


966909 


I -00 


676216 


5.43 


323784 


37 


24 | 


42894 90334 


632392 


4.43 


966849 


I -00 


676543 


5-43 


323457 


36 


25 


42920 ] 90321 


632658 


4-43 


955789 


I-OQ 


676869 


5-43 


323i3i 


35 


26 1 


42946 ; 90309 


632923 


4.43 


900729 


I -00 


677194 


5-43 


322806 


34 


27 j 


42972 90296 


633i89 


4-42 


955669 


I '00 


677520 


5-42 


322480 


33 


28 


42999 ! 90284 


633454 


4-42 


955609 


I -oo 


677846 


5-42 


322164 


32 


29 ! 


43025 90271 


633719 


4-42 


955548 


I -00 


678171 


5-42 


321829 


31 


30 


43o5i ' 90269 


633984 


4-4i 


955488 


I -00 


678496 


5-42 


32i5o4 


30 


31 j 


43077 ( 90246 


9-634249 


4-4i 


9-966428 


l-OI 


9.678821 


5-41 


10-321179 


29 


32 


43 1 04 i 90233 


6345i4 


4-4o 


9 55368 


I-OI 


679146 


5-41 


320864 


28 


33 


43i3o j 90221 


634778 


4.40 


965307 


I-OI 


679471 


5-4i 


820629 

320203 


27 


34 


43i56 90208 


635o42 


4.40 


955247 


I-OI 


679790 


5-4i 


26 


25 


43i82 j 90196 


6353o6 


4-3 9 


9 55i86 


I-OI 


680120 


5.40 


319880 


25 


36 


43209 ; 90183 


6355 7 o 


4-3 9 


9 55i26 


I-OI 


68o444 


5-40 


3l9556 


24 


37 


43235 90171 


635834 


4-3 9 


966066 


I-OI 


680768 


5-4o 


3I9232 


23 


38 


4326i i 90168 


636097 


4-38 


906006 


I-OI 


681092 


5-40 


318908 


22 


39 


43287 j 90146 


63636o 


4-38 


964944 


I -01 


6814.16 


5-3 9 


3 i 8584 


21 


40 
41 


433i3 ' 90133 


636623 


4-38 


964883 


I-OI 


681740 


5.39 


318260 


20 


4334o 90120 


9-636886 


4-3 7 


9-964823 


I-OI 


9.682063 


5.39 


10-317937 


19 


42 


43366 i 90108 


637148 


4-3 7 


964762 


I-OI 


682387 


5.39 


317613 


18 


43 


43392 ! 90096 


63741 1 


4-3 7 


954701 


I-OI 


682710 


5-38 


317290 


17 


44 


434i8 ' 90082 


637673 


4-3 7 


964640 


I-OI 


683o33 


5-38 


316967 


16 


45 


43445 ; 90070 


637935 


4-36 


964579 


I-OI 


683356 


5-38 


3i6644 


15 


46 


43471 90057 


638197 


4-36 


964618 


1-02 


683679 


5-38 


3i632i 


14 


47 


43497 ; 90045 


638458 


4-36 


954457 


1-02 


684001 


5-37 


316999 


13 


48 


43523 90032 


638720 


4-35 


964396 


1-02 


684324 


5.37 


316676 


12 


49 


43549 90019 


638981 


4-35 


954335 


1-02 


684646 


5-37 


3 1 5354 


11 


50 


43576 90007 


639242 


4-35 


964274 


I -02 
I -02 


684968 


6-37 


3i5o32 


10 


IT 


436o2 i 89994 


9-6395o3 


4-34 


9-954213 


9-680290 


5-36 


io-3i47io 


9 


52 


43628 899s 1 


63 97 64 


4.34 


964162 


1-02 


686612 


5-36 


3 1 4388 


8 


53 


43654 89968 


640024 


4-34 


964090 


1-02 


686934 


5-36 


3 1 4066 


7 


54 


4368o 89906 


640284 


4-33 


964029 


1-02 


686255 


5-36 


3i3745 


6 


55 


43706 89943 


640644 


4-33 


953968 


I -02 


6865 77 


5-35 


3i3423 


5 


56 


43733 89930 


640804 


4-33 


953906 


I -02 


686898 


5-35 


3i3io2 


4 


57 


43769 89918 


641064 


4-32 


953845 1-02 


687219 


5-35 


312781 


3 


58 


43-;?5 89906 


64i324 


4-32 


953783 


I -02 


687640 


5-35 


312460 


2 


59 


438n 89892 


641684 


4-32 


953722 


i-o3 


687861 
688182 


5.34 


3i2i39 

3n8i8 


1 


60 


4.3937 89879 


641842 


4-3i 


953660 


i-o3 


5-34 





j X. cos. N. sine. 


L. cos. 


D.l" 


L. sine. 




L. cot. | D. 1" 


L. tang. | ' 


64° 



12 



u 



SINES AND TANGENTS. 2<8 C 





Kad. = 1. 






LOGARITBMS.- 


— Radius = 10 10 . 




r 


|N. sice 


N, co& L. sine. 


D. 1" 


L. cos. 


D.1" 


L. tang. D. \" 


L. cot. 







43837 


89879 


9.641842 


4-3i 


9-953660 


i-o3 


9-688182 


5-34 


io-3>ii8i8 


60 
59 


1 


43863 


89867 


642101 


4-3i 


953599 


i-o3 


688002 


5-34 


3 1 1498 


2 


4388 9 


89854 


64236o 


4-3i 


953537 


i-o3 


688823 


5-34 


3u 177 


58 
57 


3 


43916 


89841 


642618 


4-3o 


953475 


i-o3 


689143 


5.33 


3io857 


4 


43942 


89828 


642877 


4-3o 


9534i3 


i'03 


689463 


5-33 


3 1 o537 


56 


5 


43968 


89816 


643 1 35 


4-3o 


953352 


1 -o3 


689783 


5-33 


310217 


55 


6 


43994 


89808 


6433 9 3 


4-3o 


953290 


i-o3 


690103 


5-33 


309897 


54 


7 


44020 


89790 


64365o 


4-29 


953228 


i-o3 


690423 


5-33 


309577 


53 


8 


44046 


89777 


643 9 o8 


4-29 


953i66 


i-o3 


690742 


5-32 


309258 


52 


9 


44072 


89764 


644i65 


4-29 
4-28 


953io4 


i-o3 


691062 


5-3.2 


3o8 9 38 


51 


10 
11 


44098 


89702 


644423 


953o42 


i-o3 


691381 


5-32 


308619 


50 


44124 


89739 


9-644680 


4-28 


9-952980 


1*04 


9-691700 


5-3i 


io.3o83oo 


49 


12 


44i5i 


89726 


644936 


4-28 


952918 !i-o4 


692019 


5.3i 


307981 


48 


13 


44177 


8 97 i3 


645193 


4-27 


952855 1 -04 


692338 


5-3i 


307662 


47 


14 


442o3 


89700 


64545o 


4-27 


952793 |i-o4 


692656 


5.3^ 


307.344 


46 


15 


44229 
44255 


89687 


645706 


4-27 


952731 1-04 


692975 


5-3i 


307025 


45 


16 


89674 


645962 


4-26 


952669 


1 -04 


693293 


5-3o 


306707 


44 


17 


44281 


89662 


646218 


4-26 


952606 


1-04 


693612 


5-3o 


3o6388 


43 


18 


44307 


89649 


646474 


4-26 


952044 


1-04 


698930 


5-3o 


806070 


42 


19 


44333 


8 9 636 


646729 


4-25 


952481 


1-04 


694248 


5-3o 


805752 


41 


20 
21 


4435 9 


89623 


646984 


4-2D 


952419 


i- 04 
1 -04 


694566 


5-29 


3o5434 


40 | 


44385 


89610 


9.647240 


4-25 


9.952356 


9-694883 


5-29 


io-3o5ii7 


39 


22 


444i 1 


89597 


647494 


4-24 


952294 1 -04 


69520s 


5-29 


304799 


38 


23 


44437 


8 9 584 


647749 


4-24 


95223i 1-04 


695518 


5.29 


304482 


87 


24 


44464 


8 9 5 7 i 


648004 


4-24 


902168 i-o5 


6g5836 


5.29 


304164 


36 


25 


44490 


8 9 558 


648258 


4-24 


952106 i-o5 


690153 


5-28 


3o3847 


35 \ 


26 


445 1 6 


89545 


6485i2 


4-23 


902043 ji'OOJ 


696470 


5.28 


3o353o 


34 1 


27 


44542 


89532 


648766 


4-23 


951980 |i -o5 


696787 


5-28 


3o32i3 


33 I 


28 


44568 


89519 


649020 


4-23 


951917 ji-o5 


697 1 o3 


5.28 


302897 


32 | 


29 


44594 


89506 


649274 


4-22 


c>5i854 |i.b5 


697420 


5-27 


3o258o 


31 | 


30 


44620 


8 9 4o3 


649627 


4-22 


90 1791 ji-o5 


6977^6 


5-27 


302264 


30 I 


31 


44646 


89480 


9-649781 


4-22 


9-951728 i-o5 


9-698053 


5-27 


10-301947 


29 I 


32 


44672 


89467 


65oo34 


4'22 


• 951 665 


i -o5 


698369 


5-27 


3oi63i 


28 I 


33 


44698 


89454 


650287 


4-21 


951602 


1 -oo 


698685 


5-26 


3oi3i5 


27 I 


34 


44724 


89441 


65o539 


4-21 


95i539 


i-o5 


699001 


5-26 


300999 


26 I 


85 


44720 


89428 


650792 


4-21 


951476 11 -o5 


699316 


5-26 


3oo684 


25 I 


36 


44776 


894 1 5 


601044 


4-20 


9D141 2 i-o5 


699632 


5-26 


3oo368 


24 | 


87 


44802 


89402 


65i2 97 


4-20 


901349 


1.06I 


699947 


5-26 


3ooo53 


23 1 


38 


44828 


8 9 38 9 ! 


66i549 


4- 20 


951286 


i-o6J 


700263 


5-25 


299737 


22 


39 


44854 


89376 I 


65 1 800 


4-19- 


951222 


1 -06 


700578 


5-25 


299422 


21 


40 


44880 


8 9 363 


652o52 


4-19. 


95 1 1 59 1 • 06J 


700893 


5-25 


299 1 07 


20 - 


41 


44906 


89350 


9-6523o4 


4-io 
4-18 


9-951096 ji -06! 


9-701208 


5-24 


10-298792 


19 


42 


44932 


89337 


65^555 


95io32 li -06 


701 523 


5-24 


298477 


18 


43 


44958 


89324 


6528o6 


4-18 


950968 |i-o6i 


701837 


5-24 


2 9 8i63 


17 


44 


44984 


893 1 1 


653o57 


4-i8 


950905 1 1 • 06 


702152 


5-24 


297848 


16 j 


45 


45oio 


89298 


6533o8 


4-i8 


950841 ji-o6 


702466 


5-24 


297534 


15 


46 


45o36 


89285 


653558 


4-17 


950778 li -o6! 


702780 


5-23 


297220 


14 , 


47 


45o62 


89272 


6538o8 


4-H 


950714 


1 -06! 


708095 


5-23 


296905 


13 


48 


45o88 


89259 


654o59 


4-17 


95o65o 


1 -o6J 


703409 


5-23 


296591 


12 


49 


45ii4 


89245 


654309 


4-i6 


900086 


j -06 


703723 


5-23 


296277 


11 


50 


45i4o 


89232 
89219 1 


654558 


4.16 


95o522 


l_.oq 


7o4o36 


5-22 


295964 


10 


51 


45i66 


9 -6548o8 


4-16 


9-95o458 


1-07 


9-704350 


5-22 


io- 290650 


9 


52 


45192 


89206 


655o58 


4-i6 


950394 


1-07 


704663 


5-22 


295337 


8 


53 


452i8 


89193 J 


6553o7 


4-i5 


95o33o 


1-07 


704977 


5-22 


295023 


7 


54 


45243 


89180 


655556 


4-i5 


950266 


1-07 


705290 


5-22 


2947 1 


6 


55 


45269 


89167 | 


6558o5 


4-i5 


950202 1-07! 


7o56o3 


5-21 


294397 


5 


56 


45295 


8903 | 


656o54 


4-i4 


95o 1 38 1 • 07; 


705916 


5-21 


294084 


4 


57 


45321 


89140 | 


6563o2 


4-i4 


950074 ! 1 -07I 


706228 


5-21 


293772 


3 i 


58 


45347 


89127 | 


65655i 


4-U 


950010 1 -07 


706041 


5-21 


293409 


2 J 


59 


453 7 3 


89114 


656799 


4-i3 


949945 1-07 


706854 


5-21 


293146 


1 I 


60 


45399 


89101 ! 


657047 


4-i3 


949881 1-07 


707166 


5- 20 


292834 


J 




N. 00s. 


N. sine. 


L. cos. 


D. 1" 


L. sine. | 


L. cot. 


D. 1" 


L. tsng. ' 9 


©3 G 



BINES AND TANGENTS. — 27°. 



45 



Rad. = 1. 




Logarithms.— 


Radius^ 10 10 . 







N. sine. ; 

45399 ! 


N. cos.1 


L. sine. 


D. 1" 


L. cos. 


D.l" 


L. tang. 


D.l" 


L. cot. 




89101 


9-657047 


4-i3 


9-949881 


1-07 


9.707I66 


5-20 


10-292834 


60 


1 


45425 ; 89087 


657295 


4-i3 


949816 


I.07 


707478 


5-20 


292522 


59 


2 


4D45 1 j 89074 


657542 


4-12 


949732 


1-07 


707790 


5-20 


292210 


58 


3 


45477 ' 89061 


637790 


4-12 


949688 


1-08 


708102 


5-20 


291898 


57 


4 


455o3 ! 89048 


658o37 


4-12 


949623 


I- 08 


708414 


5.19 


29086 


56 


5 


45529 ; 89035 


658284 


4-12 


949558 


i- 08 


708726 


5-i 9 


291274 


55 


6 


45554 89021 


65853i 


4-n 


949494 


1-08 


709037 


5-i 9 


290963 


54 


7 


45580 ; 89008 


658778 


4-u 


949429 


1-08 


709349 


D-I 9 


290651 


53 


8 


456o6 i 88995 


659025 


4-n 


949364 


1-08 


709660 


5.19 


290340 


52 


9 


45632 i 88981 


659271 


4-io 


949300 


1-08 


709971 


5.18 


290029 


51 


10 


45658 88968 


659517 


4-io 


949235 


1.08 


710282 


5.18 


289718 


50 


11 


45684 : 88955 


9-659763 


4-io 


9.949170 


i- 08 


9-710593 


5. 18 


10-289407 


49 


12 


45710 1 88942 


660009 


4.09 


949105 


1-08 


710904 


5.18 


289096 


48 


13 


45736 | 88928 


66o255 


4.09 


949040 


1-08 


71 1 2 ID 


5-i8 


288785 


47 


14 


40762 88915 


66o5oi 


4.09 


948975 


1-08 


71 i525 


5.17 


288475 


46 


15 


45787 88902 


660746 


4.09 
4-08 


948910 


i- 08 


7ii836 


5-17 


288164 


45 


16 


458 1 3 88888 


660991 


948845 


i- 08 


712146 


5-17 


287854 


44 


17 


45839 88875 


661236 


4-08 


948780 


1-09 


712456 


5-17 


287D44 


43 


18 


45865 88862 


661481 


4-o8 


948715 


1-09 


712766 


5-i6 


287234 


42 


19 


40891 I 88848 


661726 


4-07 


94865o 


1-09 


713076 


5- 16 


286924 


41 


20 
21 


45917 88835 


661970 


4-0,7 


948584 


1-09 


7i3386 


5-i6 


286614 


40 


45942 [88822 


9-662214 


4-07 


9-948519 


1 -09 


9-713696 


5-i6 


io-2863o4 


39 


22 


43968 88808 


662459 


4-07 


948454 


1-09 


714005 


5-i6 


285995 


38 


23 


45994 


88795 


66270J 


4. 06 


948388 


1-09 


7I43U 


5-i5 


285686 


37 


24 


46020 


88782 


662946 


4- 06 


948323 


1-09 


714624 


5-i5 


285376 


36 


25 


46046 88768 


663190 


4-o6 


948257 


1.09 


714933 


5-i5 


285067 


35 


26 


46072 I 88755 


663433 


4-o5 


948192 


1-09 


715242 


5-i5 


284758 


34 


27 


46097 


88741 


663677 


4-o5 


948126 


1-09 


7i555i 


5-14 


284449 


33 


28 


46123 


88728 


663920 


4-o5 


948060 


1.09 


7i586o 


5-i4 


284140 


32 


29 


46149 


88715 


664i63 


4-o5 


94799 5 


I-IO 


716168 


5-14 


283832 


31 


30 
31 


46173 188701 


664406 


4.04 


947929 


I'lO 
I-IO 


716477 


5-i4 


283523 


30 


46201 i 88688 


9.664648 


4.04 


9-947863 


9-716785 


5-i4 


io-2832i5 


29 


32 


46226 | 88674 


664891 


4-04 


947797 


I -10 


717093 


5-i3 


282907 


28 


33 


46252 88661 


665i33 


4-o3 


94773i 


I-IO 


717401 


5-i3 


282599 


27 


34 


46278 88647 


665375 


4-o3 


947665 


I-IO 


717709 


5-i3 


2S2291 


26 


35 


463 04 


88634 


665617 


4-o3 


947600 


I-IO 


718017 


5-i3 


281983 


25 


36 


4633o 


88620 


66585 9 


4-02 


947533 


I-IO 


7i8325 


5-i3 


281675 


24 


37 


46355 


88607 


666100 


4-02 


947467 


I-IO 


7i8633 


5-12 


28l367 


23 


38 


463 8 1 


885g3 


666342 


4-02 


9474oi 


I-IO 


718940 


5-12 


281060 


22 


39 


46407 { 8858o 


666583 


4-02 


947335 


I -10 


719248 


5-12 


280752 


21 


40 
41 


46433 88566 


666824 


4-01 


947269 


I-IO 


719555 


5-12 


280445 


20 


46458 ! 88553 


9-667065 


4-oi 


9.947203 


I -10 


9-719862 


5.12 


io-28oi38 


19 


42 


46484 88539 


6673o5 


4-oi 


947i36 


I-II 


720169 


5-n 


279831 


18 


48 


465 10 | 88526 


667046 


4-oi 


947070 


I'll 


720476 


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I'M 


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9 


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1 


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671609 


3.96 


945935 


I-I2 


725674 


5-o8 


274326 







N. cos. 


N. sine. 


L. cos. 


D.l" 


1 L. sine. 




L. cot. 


D.l" 


L. tang. 


' 


62° 



46 



SINES AND TANGENTS. — 28 c 



Rad. = 1. 


Logarithms. — Radius = 10 10 . 


' 


N. sine J N.cos. 


L. sine. 


D. \" 


L. cos. 


D.l" 


L. tang. 


D.l" 


L. cot. 







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48 


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40 


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10-267952 


89 


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265537 


31 


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31 


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47741 


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87868 


678663 


3 


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i-i4 


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9 -735o66 


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29 


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28 


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11 


50 
51 


48226 


87603 


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3 


82 


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740767 


4.98 


25 9 233 


10 


48252 


87589 

8 7 5 7 5 


9-6835i4 


3 


82 


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9.741066 


4.98 


10-258934 


9 


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74i365 


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258635 


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53 


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683972 


3 


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741664 


4.98 


258336 


7 


54 


48328 


87546 


684201 


3 


81 


942239 


1-16 


741962 


4*97 


258o38 


6 


55 


48354 


87532 


68443o 


3 


81 


942169 


1-16 


742261 


4-97 


257739 


5 


56 


48379 


8 7 5i8 


684658 


3 


81 


942099 


1-16 


742559 
742858 


4-97 


257441 


4 


57 


484o5 


87504 


684887 


3 


80 


942029 


1-16 


4-97 


257142 


3 


58 


4843o 


87490 


685n5 


3 


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941959 


1-16 


743 1 56 


4-97 


256844 


2 


59 


48456 


87476 


685343 


3 


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941889 


1-17 


743454 


4-97 


256546 


1 


60 


48481 


87462 


685571 


3 


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941819 


1-17 


743752 


4.96 


256248 







N. cos. 


N. sine. 


L. cos. 


D 


1" 


L. sine. 




L. cot. 


D.l" 


L. tang. 


' 


61° 



SINES AND TANGENTS. 29°. 



47 



Rad. = 1. 


Logarithms. — Radius = 10 10 . 


' 


N.sine. 


N. cos. 


L. sine. 


D 


1" 


L. cos. 


D.1" 


L. tang. 


D. 1" 


L. cot. 






1 

2 
3 

4 
5 
6 
7 
8 
9 
10 

11 
12 
13 
14 
15 
16 
17 
18 
19 
20 


48481 
485o6 
48532 
48557 
48583 
48608 
48634 
48659 
48684 
48710 
48735 


87462 
87448 
87434 
87420 
87406 
87391 
87377 
8 7 363 

87349 
87335 
87321 


9-685571 
685799 
686027 
686254 
686482 
686709 
686g36 
687163 
687389 
687616 
687843 


3 

3 
3 
3 
3 
3 
3 
3 
3 
3 
3 


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•79 
•79 
•79 
•79 

"■2 

78 
77 
77 


9-941819 
941749 
941679 
941609 
94i539 
941469 
941398 
9 4i328 
941258 
941187 
941 1 17 


1-17 
i- 17 
1-17 
1. 17 
1. 17 
1. 17 
I- 1 7. 
i- 17 

i-i-7 

1-17 

I'll 


9-743752 
744o5o 
744348 
744645 
744943 
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745538 
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746i32 
746429 
746726 


4-96 
4-96 
4.96 
4.96 
4-96 
4-96 
4-95 
4-95 
4-95 
4-9 5 
4-95 


10-256248 
266960 
255652 
255355 
255o57 
254760 
264462 
254i65 
253868 
253671 
253274 


60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 


48761 
48786 
488 1 1 
4883 7 
48862 
48888 
48913 
48 9 38 
48964 
48989 


87306 
87292 
87278 
87264 
87250 
87235 
87221 
87207 
87193 
87,78 


9-688069 
688295 
688521 
688747 
688972 
689198 
689423 
689648 
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690098 


3 
3 
3 
3 
3 
3 
3 
3 
3 
3 


77 
77 

76 

76 

7 ^ 
7 5 
75 


9-941046 
940975 
940905 
940834 
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940693 
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940480 
940409 


1-18 
1-18 
1-18 
i-i8 
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i- 18 
1. 18 
1-18 
1-18 
1-18 


9-747023 
747319 
747616 
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748209 
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748801 
749097 
749393 
749689 


4.94 
4.94 
4.94 
4.94 
4.94 
4-93 
4.93 
4-93 
4-93 
4- 9 3 


10-262977 
25268i 
252384 
262087 
261791 
261495 
25i 199 
250903 
250607 
25o3n 


49 
48 
47 
46 
45 
44 
43 
42 
41 
40 


21 
22 
23 
24 
25 
26 
27 
28 
29 
30 


49014 
49040 
49065 
49090 
49116 
49U1 
4yi66 
4919 2 
49217 
49242 


87164 
87100 
87136 
87121 
87107 
87093 
87079 
87064 
87050 
87036 


9-690323 
690548 
690772 
690996 
691220 
691444 
691668 
691892 
6921 1 5 
692339 


3 
3 
3 
3 
3 
3 
3 
3 
3 
3 


74 
74 
74 
74 
73 
73 
73 
72 
72 
72 


9«94o338 
940267 
940196 
940125 
940054 
939982 
9399 1 1 
939840 
939768 
939697 


1. 18 

1. 18 
1-18 

1. 19 
1. 19 

1*19 
1-19 
1. 19 

1*19 
1. 19 


9 • 749985 
750281 
750576 
750872 
761167 
751462 
75i757 
752o52 
752347 
752642 


4-93 
4-92 
4-92 
4-92 
4-92 
4-92 
4-92 
4-91 
4-91 
4-91 


io-25ooi5 

249719 
249424 
249128 
248833 
248538 
248243 
247948 
247653 
247358 


39 
38 
37 
36 
35 
34 
33 
32 
31 
30 


31 
32 
33 
34 
85 
86 
37 
38 
39 
40 

41 
42 
43 
44 
45 
46 
47 
48 
49 
50 


49268 
49293 
493i8 
49344 
49369 
49394 
49419 
49445 
49470 
49495 


87021 
87007 
86993 
86978 
86964 
86949 
86 9 35 
86921 
86906 
86892 


9-692562 
692785 
693008 
693231 
693453 
693676 
6 9 38 9 8 
694120 
694342 
694564 


3 
3 
3 
3 
3 
3 
3 
3 
3 
3 


72 
7i 
li 
7i 
7i 
70 
70 
70 
70 
69 


9-939625 
939D54 
939482 
939410 
939339 
939267 
939195 
939123 
939052 
938980 


1-19 
1. 19 
1. 19 
1. 19 
I- 19 

1-20 
1-20 
1-20 
1-20 
1-20 


9-752937 
75323i 
753526 
753820 
7541 i5 
754409 
754703 

754997 
755291 
755585 


4*91 
4-91 
4*91 
4.90 
4-90 
4.90 
4.90 
4.90 
4.90 
4-8 9 


10-247063 
246769 
246474 
246180 
245885 
245691 
246297 
245oo3 
244709 
2444 1 5 


29 
28 
27 
26 
25 
24 
23 
22 
21 
20 


49521 
49546 

49571 
49596 
49622 
49647 
49672 
49697 
49723 
49748 


86878 
86863 
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86820 
868o5 
86791 
86777 
86762 
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9-694786 
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695229 
695450 
690671 
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696 1 1 3 
696334 
696554 
696775 


3 
3 
3 
3 

3- 
3 
3- 
3- 
3- 
3 


t 9 

A 9 
69 

68 
68 
68 
68 
67 

67 
67 


9-938908 
938836 
938763 
938691 
938619 
938547 
938475 
938402 
93833o 
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I -20 
1-20 
1-20 
I -20 
1-20 
1-20 
1-20 
I-2I 
"-•21 
"'•21 


9.755878 
756172 
756465 
756759 
757052 
757345 
757638 
75793i 
758224 
758617 


4.89 
4-8 9 
4.89 
4-8 9 
4-89 
4-88 
4-88 
4-88 
4-88 
4-88 


10-244122 

243828 
243535 
243241 
242948 
242666 
242362 
242069 
241776 
241483 


19 

18 
17 
16 
15 
14 
13 
12 
11 
10 


51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


49773 
49798 
49824 
49849 
49874 
49899 
49924 
49900 
49975 
Soooo 


86733 
86719 
86704 
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866 7 5 
86661 
86646 
86632 
86617 
866o3 


9-696995 
697215 
697435 
697654 
697874 
698094 
6 9 83 1 3 
698532 
698751 
698970 


3- 

3 
3 
3 
3 
3 
3 
3- 
3 
3 


67 
66 
66 
66 
66 
65 
65 
65 
65 
64 


9-938183 

9 38n3 
938040 
937967 
937895 
937822 
937749 
937676 
937604 
937531 


-•21 
I-2I 
'■•21 
I - 21 
[•21 
-•21 
I -21 
I-2I 
I-2I 
1 -21 


9-758810 
769102 
769395 
759687 

759979 
760272 
760664 
76o856 
761148 
761439 


4-88 
4-8 7 
4-8 7 
4-8 7 
4-8 7 
4-8 7 
4-8 7 
4-86 
4-86 
4-86 


10-241 190 
240898 

24o6o5 
24o3i3 
240021 
239728 
239436 
23gi44 
238852 
23856i 


9 
3 
7 
6 
5 
4 
3 
2 
1 



1 


N. cos. 


N. sine.] L. cos. 


D. 1" j L. sine. 




L. cot. 


D.l' 


L. tang. 


> 


60° 



4:8 



SINES AND TANGENTS. — 30°. 



Ead. = 1. 




IiOGARITHMS.- 


-Radius = 10 10 . 




' 


N. sine. 


N. cos. 


L. sine. 


D. 1" 


L. COS. 


D.l" 


L. tang. 


Dl." 


L. cot. 







5oooo 


866o3 


9-698970 
699189 


3-64 


9-93753l 


I-2I 


9-761439 


4-86 


io- 23856i 


60 


1 


5oo25 


86588 


3-64 


937458 


1-22 


761731 


4-86 


238269 


59 


2 


5oo5o 


86573 


699407 


3-64 


937385 


1-22 


762023 


4-86 


237977 


58 


3 


50076 


8655 9 


699626 


3-64 


937312 


1-22 


762314 


4-86 


237686 


57 


4 


5oioi 


86544 


699844 


3-63 


937238 


1-22 


762606 


4-85 


237394 


56 


5 


5oi26 


8653o 


700062 


3-63 


937165 


I -22 


762897 


4-85 


237103 


55 


6 


5oi5i 


865 1 5 


700280 


3-63 


937092 


I -22 


763i88 


4-85 


236812 


54 


7 


50176 


865oi 


700498 


3-63 


937019 


I -22 


763479 


4-85 


236521 


53 


8 


50201 


86486 


700716 


3-63 


936946 
936872 


1-22 


763770 


4-85 


23623o 


52 


9 


50227 


86471 


700933 


3-62 


1-22 


764061 


4-85 


235939 


51 


10 
11 


90252 


86457 


70ii5i 


3-62 


936799 


1-22 


764352 


4-84 


235648 


50 
49 


50277 


86442 


9'70i368 


3-62 


9-936725 


1-22 


9-764643 


4-84 


10-235357 


12 


5o3o2 


86427 


701 585 


3-62 


936652 


1-23 


764933 


4-84 


235067 


48 


13 


50327 


864i3 


701802 


3-6i 


936578 


1-23 


765224 


4-84 


234776 


47 


14 


5o352 


863 9 8 


702019 


3-6i 


9365o5 


1-23 


7655i4 


4-84 


234486 


46 


15 


5o377 


86384 


702236 


3- 61 


93643i 


1-23 


7658o5 


4-84 


234195 


45 


16 


5o4o3 


8636 9 


702452 


3-6i 


936357 


1-23 


766095 


4-84 


233905 


44 


17 


5o428 


86354 


702669 
702885 


3-6o 


936284 


1-23 


766385 


4-83 


2336i5 


43 


18 


5o453 


8634o 


3-6o 


936210 


1-23 


766675 


4-83 


233325 


42 


19 


50478 


86325 


7o3ioi 


3- 60 


936i36 


1-23 


766965 


4-83 


233o35 


41 


20 


5o5o3 


863 10 


703317 


3-6o 


936062 


1-23 


767255 


4-83 


232745 


40 


21 


5o528 


86295 


9 -7o3533 


3.59 


9-935988 


1-23 


9-767545 


4-83 


10-232455 


39 


22 


5o553 


86281 


703749 


3-5 9 


935914 


1-23 


767834 


4-83 


232i66 


38 


23 


5o5 7 8 


86266 


703964 


3-5 9 


935840 


1-23 


768124 


4-82 


231876 


37 


24. 


5o6o3 


8625i 


704179 


3-5 9 


935766 


1-24 


768413 


4-82 


23i587 


36 


25 


50628 


86237 


704395 


3-5 9 
3-58 


935692 


1-24 


768703 


4-82 


231297 


35 


26 


5o654 


86222 


704610 


9356i8 


1.24 


768992 
769281 


4-82 


23ioo8 


34 


27 


50679 


86207 


704825 


3-58 


935543 


1-24 


4-82 


230719 


33 


28 


50704 


86192 


706040 


3-58 


935469 
935396 


1-24 


769570 


4-82 


23o43o 


32 


29 


5o 7 2 9 


86178 


7o5254 


3-58 


1-24 


769860 


4-8i 


23oi4o 


31 


30 


50754 


86 1 63 


705469 


3-57 


935320 


1-24 


770U8 


4-8i 


229852 


30 


31 


5 z°l 19 


86148 


9-7o5683 


3-5 7 


9-935246 


1-24 


9-770437 


4-8i 


10-229363 


29 


32 


5o8o4 


86i33 


705898 


3.57 


935171 


1-24 


770726 


4-8i 


229274 


28 


83 


50829 


86119 


7061 1 2 


3.57 


935097 


1-24 


77ioi5 


4-8i 


228985 


27 


34 


5o854 


86104 


706326 


3-56 


935o22 


1-24 


77i3o3 


4-8i 


228697 


26 


35 


50879 


86089 


706539 
706753 


3-56 


934Q48 


1-24 


771592 


4-8i 


228408 


25 


36 


50904 


86074 


3-56 


934873 


1-24 


771880 


4-8o 


228120 


24 


37 


50929 


86o5 9 
86o45 


706967 


3-56 


934798 


1-25 


772168 


4-8o 


227832 


23 


38 


D0954 


707180 


3-55 


934723 


1-25 


772457 


4-8o 


227543 / 


22 


39 


50979 


86o3o 


707393' 


3-55 


934649 


1-25 


772745 


4-8o 


227255 


21 


40 


5ioo4 


860 1 5 


707606 


3-55 


934574 


1-25 


773o33 


4-8o 


226967 


20 


41 


51029 


86000 


9-707819 


3-55 


9-934499 


1-25 


9-773321 


4- 80 


10-226679 


19 


42 


5io54 


85 9 85 


708032 


3-54 


934424 


1-25 


773608 


4-79 


226392 


18 


43 


51079 


85970 


708245 


3-54 


934349 


1-25 


773896 
774184 


4-79 


226104 


17 


44 


5no4 


85 9 56 


708458 


3-54 


934274 


1-25 


4-79 


2258i6 


16 


45 


51129 


85 9 4i 


708670 


3-54 


934199 


I -25 


774471 


4-79 


223529 


15 


46 


5u54 


85 92 6 


708882 


3-53 


934123 


1-25 


774759 


4-79 


225241 


14 


47 


5ii79 


85 9 n 


709094 


3-53 


934048 


1-25 


775046 


4-79 


224934 


13 


48 


5 1 204 


858o6 


709306 


3-53 


933973 
933898 


1-25 


775333 


4-79 


224667 


12 


49 


01229 


8588i 


709518 


3-53 


1-26 


775621 


4-78 


224379 


11 


50 
51 


5i254 


85866 


709730 


3-53 


933822 


1-26 


775908 


4-78 


224092 


10 


5i2 79 


8585i 


9-70994I 


3-52 


9.933747 


1-26 


9-776195 


4-78 


io-2238o5 


9 


52 


5i3o4 


85836 


7ioi53 


3-52 


933671 


1-26 


776482 


4-78 


2235i8 


8 


53 


5i329 


85821 


7io364 


3-52 


933596 


1-26 


776769 


4-78 


22323l 


7 


54 


5 1 354 


858o6 


710573 


3-52 


933520 


1-26 


777055 


4.78 


222945 


6 


55 


51379 


85792 


710786 


3-5i 


933445 


1-26 


777342 


4-78 


222658 


5 


56 


5 1 404 


85 777 


710997 


3-5i 


933369 
933293 


1-26 


777628 


4-77 


222372 


4 


57 


51429 


85762 


711208 


3-5i 


1-26 


777915 


4-77 


222085 


3 


58 


5 1 454 


85 7 47 


71U19 


3-5i 


933217 


1-26 


778201 


4-77 


221799 


2 


59 


5 1 479 


85 7 32 


711629 


3-5o 


933i4i 


1-26 


778487 


4-77 


22l5l2 


1 


60 


5i5o4 


83717 


71 1839 


3-5o 


933o66 


1-26 


778774 


4-77 


221226 





N. cos. N. sine. 


L. cos. 


D.l" 


L. sine. 




L. cot. 


D.l" 


L. tang. 


' 






59° 







SINES AND TANGENTS. 



-$i c 



49 



Rad. = 1. 




Logarithms. — 


Radius = 10'°.. 




/ 


N.siae. 


N. cos. 


L. sine. 


D. 1" 


L. cos. I 


"i 


L. tang. 


D. V 


L. cot. 







5i5o4 


85 7 i 7 


9-711839 


3-5o 


9-933066 i 


■26; 


9-778774 


i-Ti 


10-221226 


60 


1 1 


5i529 


85702 


7i2o5o 


3-5o 


932990 i 


•27 


779060 


4.77 


220940 


59 


; 2 


5 1 554 


85687 


712260 


3.5o 


932914 1 


•27j 

.27 


779346 


4-76 


22o654 


58 


• 3 


5i579 


85672 


712469 


3-49 


9 32838 1 


779632 


4.76 


220368 


57 


4 


5 1 604 


8565 7 


712679 


3-49 


932762 1 


.27! 


779918 


4.76 


220082 


56 


5 


5i628 


85642 


712889 


3-49 


9 32685 1 


•271 


780203 


4-76 


219797 


55 


6 


5*653 


8562 7 


713098 


3-49 


932609 1 


•27 


780489 


4-76 


219511 


54 


i s 


51678 


856 1 2 


7i33o8 


3-49 


932533 1 


.27 


780775 


4-76 


219225 


53 


51703 


855 97 


7 1 3 5 1 7 


3-48 


932457 1 


•27 


781060 


4-76 


218940 


52 


' 9 


01728 


85582 


713726 


3-48 


93238© 1 


•27 


781346 


4-75 


2186D4 


51 


10 
11 


5i 7 53 


8556 7 


713935 


3-48 


9323o4 1 


.27 


78i63i 


4-75 


2i836 9 


50 


5i 77 8 


8555i 


•9-7i4i44 


3-48 


9-932228 1 


•27' 


9-781916 


4-75 


10-218084 


49 


12 


5i8o3 


85536 


714352 


3-47 


932i5i 1 


.27 


782201 


4-73 


217799 


48 


13 


51828 


85521 


7 i 456 1 


3-47 


932075 1 


.28 
.28 


782486 


4-75 


217514 


47 


14 


5i852 


855o6 


714769 


3-47 


931998 1 


782771 


4-7§ 


217229 


46 


15 


5i8 77 


85491 


714978 


3-47 


031921 1 


.28 


783o56 


4-7^ 


216944 


45 


it 


51902 


85476 


71 5 1 86 


3-47 ; 


9 3i845 1 


-28! 


783341 


4-7-5 


216659 


44 


17 


5i 9 2 7 


8546i 


715394 


3-46 


931768 1 


-28 


783626 


4-74 


216374 


43 


18 


51952 


85446 


7i56o2 


3-46 


931691 1 


.28 


783910 


4-74 


216090 


42 


19 


5i 977 


8543 1 


715809 


3-46 


931614' 1 


.28 


784195 


4-74 


2i58o5 


41 


20 


5 2002 


854i 6 


716017 


3-46 


93i537 1 


•28 


784479 


4-74 


2l552I 


40 


21 
22 


52026 


85401 


9-716224 


3-45 


9-931460 i 


• 28 


9-784^64 


4-74 


io-2i5236 


39 


5so5i 


85385 


7*6432 


3-45 


9 3i383 1 


.28 


78D048 


4-74 


214952 


38 


23 


52076 


85370 


716639 


3-45 


93i3o6 1 


.28 


785332 


4-73 


214668 


37 


24 


52I0I 


85355 


716846 


3-45 


931229 1 


• 29 


7856i6 


4-73 


214384 


36 


25 
26 


52126 


8534o 


717053 


3-45 


93u52 1 


■29 


785900 


4-73 


214100 


35 


52i5i 


85325 


717259 


3-44 


931075 1 


.29 


786184 


4-73 


2i38i6 


34 


27 


52175 


853io 


717466 


3-44 


930998 1 


• 29 


786468 


4-73 


2i3532 


33 


23 


52200 


85294 


717673 


3-44 


930921 i 


.29 


786752 


4-73 


2i3248 


32 


29 


52225 


85279 


717879 


3-44 


93o843 i 


•29 


787036 


4-73 


'212964 


31 


30 

31 


52250 


85264 


718085 


3-43 
3-43 


930766 i 


•29 1 


787319 


4-72 


212681 


30 
29 


52275 


85249 


9-718291 


9.930688 i 


•29 


9.787603 


4.72 


10-212397 


32 


52299 


85a34 


718497 


3-43 


93061 1 ] 


.29 


787886 


4- 72 


212114 


28 


33 


52324 


852 1 8 


718703 


3-43 


93o5J3 


•29 


788170 


4-72 


2ii83o 


27 


; 34 


52349 


852o3 


718909 


3-43 


93o456 3 


•29 


788453 


4-72 


2ii547 


26 


| 35 


52374 


85i88 


719114 


3-42 


930378 


•29 


7 88 7 36 


4-72 


21 1264 


25 


■ 36 


52399 


85i73 


719320 


3-42 


93o3oo 


• 3o 


789019 


4-72 


210981 


24 


: 37 


52423 


85i5 7 


719525 


3-42 


93o223 


-3o 


789302 


4-71 


210698 


23 


; 38 


52448 


85 1 42 


719730 


3'- 42 


93oi45 


-3o 


789585 


4-71 


2io4i5 


22 


; 39 


52473 


85i?7 


719935 


3-4i 


930067 


-3o 


789868 


4-7i 


2IOI32 


21 


| 40 

1 41 

J 42 


52498 


85ii2 


720140 


3-4i 


929989 


-3o 


7901 5 1 


4-7i 


209849 


20 


52322 


85o 9 6 


9-720345 


3-4i 


9-929911 


-3o 


9-790433 


4-71 


10-209567 


19 


52547 


85o8i 


720549 


3-4i 


929833 


-3o 


790716 


4-71 


209284 


18 


f 43 


52572 


85o66 


720754 


3-4o 


929755 


-3o 


790999 
791281 


4-7i 


209001 


17 


f 44 


52597 


■85o5i 


720958 


3-40 


929677 


[-3o 


4-7* 


208719 


16 


I 45 


52621 


85o35 


721162 


3-4o 


929599 


•3o 


79 1 563 


4-70 


208437 


15 


i * 6 


52646 


85o2o 


72i366 


3-40 


929521 


-3o 


791846 


4-70 


208154 


14 


1 47 


52671 


85oo5 


721570 


3-4o 


929442 


-3o 


792128 


4-7<> 


207872 


13 


1 48 


52696 


84989 


721774 


3-3 9 


929364 


-3i 


792410 


4-70 


207590 


12 


I 49 


52720 


84974 


721978 


3-3 9 


929286 


[•3i 


792692 


4.70 


207308 


11 


1 ^ 


52745 


84959 


722181 


3-3 9 


929207 


■ 3i 


792974 


4-7° 


207026 


10 


1 51 


52770 


84943 


•9-722385 


p 9 " 


9-929129 


[-3i 


9-793206 


4-70 


10-206744 


9 


1 52 


52794 


84928 


'722588 


3-3 9 


929050 
928972 


[•3i 


793538 


4-69 


206462 


8 


1 5S 


52819 


84913 


722791 


3.38 


i-3i 


793819 


4-69 


20618* 


7 


| 54 


52844 


84897 
84882 


722994 


3-38 


928893 


r-3i 


7941 01 


4-69 


205899 


6 


1 55 


52869 


723197 


3-38 


928815 


-3 1 


794383 


4-69 


205617 


5 


1 5* 


52893 


84866 


723400 


3-38 


928736 


■ 3i 


794664 


4-69 


2o5336 


4 


1 »7 


52918 84851 


7236o3 


3-37 


928657 


-3« 


794945 


4-69 


2o5o55 


3 


? .58 


52943 ! 84836 


7238o5 


3-37 


928578 


•3i 


795227 


4-6q 


204773 


2 


I 59 


52967 84820 


724007 


3-3 7 


928499 


-3j 


795508 


4-68 


204492 


1 


1 *° 


52992 ! 84805 


724210 


3-3 7 


928420 


•3i 


! 795789 1 4-68 


2042 1 1 





1 


N, cos. N. sine. 


j L. cos. 


D.l" 


L. sine. 




L.00I D. 1" 


L. tang. 


' 


»■*'■ ' . 




58° 









50 



SINES AND TANGENTS. 32' 



Kad. = 1. 




LOGARITHMS.- 


—Radius — 10'°'. 




r 


N-sme. 


N. cos 


L, sloe: 


D.l" 


L. cos. 


D.l" 
r-32 


L. tang. 


ft* 


L. cot. 







62992 


84805 


9.724210 


3-3 7 


9-928420 


9-795789 


4-68 


to- 20421 1 


60 


1 


53oi7 


84789 


724412 


3.37 


928342 


1-32 


796070 


4-68 


203930 


59 


2 


53o4r 


84774 


724614 


3-36 


928263 


1-32 


79635*1 


4-68 


208649 


58 


3 


53o66 


84759 


724816 


3-36 


928183 


1-32 


796632 


4-68 


2o3368 


il 


4 


53091 


84743 


720017 


3-36 


928104 


1-32 


796913 


4-68 


203087 


5 


53u5 


84728 


725219 


3-36 


928025 


J- 32 


797194 


4-68 


202806 


55 


6 


53 1 40 


84712 


725420 


3-35 


927946 


1-32 


797473 


4-68 


202020 


54 


7 


53i64 


84697 


725622 


3-35 


927867 


1-32 


797750 


4-68 


202-240 


53 


8 


53189 


84681 


725823 


3-35 


927787 


1-32 


798036 


4-67 


201964 | 52 


9 


53214 


84666 


726024 


3-35 


927708 


1-32 


798316 


4-67 


201684 


51 


10 
11 


53238 


846 5o 


726225 


3-35 


927629 


1-3? 


798596 


4-67 
4-67 


201404 


50 


53263 


84635 


9-726426 


3-34 


9-927349 


1-32 


9.798877 


10-201123 


49 


12 


53288 


84619 


726626 


3-34 


927470 


i-33 


799107 


4.67 


200843 


48 


18 


533 1? 


84604 


726827 


3-34 


927390 


i-33 


799437 


4.67 


20o563 


47 
46 


14 


5333 7 


84588 


727021 
72722b 


3-34 


927310 


1-33 


799717 


4-67 


200283 


15 


5336 1 


84573 


3-34 


927231 


i-33 


799997 


4-66 


2O0003 


45 


16 


53386 


84557 


727428 


3-33 


927151 


i-33 


800277 


4-66 


199723 


44 


17 


534ii 


84542 


727628 


3-33 


927071 


i-33 


800557 


4-66 


199443. 


43 


18 


53435 


84526 


727828 


3.33 


.926991 


i-33 


8oo836 


4-66 


1 99 164 


42 


19 


5346o 


845 1 1 


728027 


3-33 


92691 1 
92683 r 


1-33 


801116 


4-66 


198884 


41 


20 
21 


53484 


84495 


728227 


3-33 


1-33 


801396 


4-66 


1 98604 


40 
39 


535og 


84480 


9-728427 


3-32 


9-926751 


1-33 


9-801675 


4-66 


ro- 198325 


22 


53534 


84464 


728626 


3-32 


92667 1 


i-33 


801955 


4-66 


1 98045 


38 


23 


53558 


84448 


728825 


3-32 


926591 


i-33 


802234 


4-65 


197766 


37 


24 


53583 


84433 


729024 


3-32 


9265ii 


1-34 


8o25i3 


4-65 


197487 


36 


25 


53607 


84417 


729223 


3-3i 


926431 


1-34 


802792 


4-65 


197208 


35 


26 


53632 


84402 


729422 


3-3r 


92635? 


1 -34 


803072 


4-65 


196928 


34 


27 


53656 


84386 


729621 


3-3i 


926270 


1-34 


8o335i 


4-65 


j 96649. 


83 


28 


5368 1 


84370 


729820 


3-3? 


926190 


1-34 


8o363o 


4-65 


196370 


32 


29 


53 7 o5 


84355 


730018 


3-3o 


9261 10 


1-34 


803908 


4-65 


196092 


31 


30 
31 


5373o 


84339 


730216 


3-3o 


926029 


1-34 


804187 


4-65 


190813 


30 
'29 


53754 


84324 


9-73o4i5 


3-3o 


9-925949 


j- 34 


9.-804466 


4.64 


ro- 195534 


32 


53779 


843o8 


73o6i3 


3-3o 


925868 


1-34 


8o4745 


4-64 


i'95255 


28 i 


33 


538o4 


84292 


73o8i 1 


3-3o 


925788 


1-34 


8o5oq3 


4-64 


194977 


27 


34 


53828 


84277 


731009 


3- 29 


925707 


1-34 


8o53o2 


4-64 


194698 


26 


85 


53853 


84261 


731206 


3-29 


925626 


1-34 


8o558o 


4-64 


194420 


25 


36 


53877 


84245 


731404 


3-29 


925545 


i-35 


8o585 9 


4-64 


194141 


24 | 


37 


53902 


8423o 


7^1602 


3-29 


9^5465 


i-35 


806137 


4-64 


1 9 3863 


23 | 


88 


53926 


84214 


731799 


3- 29 

3-28 


925384 


1-35 


80641 5 


4-63 


193585 


22 i 


89 


5395f 


84198 
84182 


731996 


9253o3 


1-35 


8o66 9 3 


4-63 


1 93307 


21 I 


40 


53975 


732193 


3-28 


925222 


i-35 


80697 j 


4-63. 


193029 


20 


41 


54000 


84167 


9- 7323oo 


3-28 


9-925141 


i-35 


9-807249 


4-63 


10-192751 


19 | 


42 


54024 


84i5i 


7 3258 7 


3.28 


925o6o 


r-35 


807527 


4-63 


192473 


18 


43 


54049 


84i35 


732784 


3-28 


924979 
924897 


r-35 


807805 


4-63 


192190 


17 : 


44 


54073 


84120 


732980 


3-27 


i-35 


8o8o83 


4-63 


191917 


16 i 


45 


54097 


84104 


733177 


3-27 


924816 


r-35 


8o836i 


4-63 


1 91639 


15 | 


46 


54122 


84088 


7 333 7 3 


3.27 


924735 


r-36 


8o8638 


4-62 


1 91 362 


14 ! 


17 


54146 


84072 


733569 


3.27 


924654 


i.36 


808916 


4-62 


1 91-084 


13 1 


48 


54171 


840D7 


733 7 65 


3-27 


924572 


i-36 


809193 


4-62 


190807 


12 


49 


54195 


84041 


733961 


3-26 


924491 


i-36 


80947 j 


4-62 


1 900 29. 


11 


50 


54220 


84025 


734i57 


3-26 


924409 


i-36 


809748 


4-62 


190252 


10 j 


51 


54244 


84009 


9-734353 


3-26 


9.924328 


1-36 


9-810025 


4-62 


so- 189975 


9 ! 


52 


54260 
54 29 J 


8 3 99 4 


734549 


3-26 


924246 


?-36 


8io3oi 


4-62 


189698 


8- 


53 


83 97 8 


734744 


3-25 


924164 


1-36 


8io58o 


4-62 


189420 


7 ! 


54 


543i 7 


83 9 62 


734939 


3-25 


924083 


i-36 


810857 


4-62 


189143 
1,88866 


6 


55 


54342 


83946 


735i35 


3-25 


924001 


1-36 


8lii34 


4-6? 


5 


56 


54366 


83g3o 


73533o 


3-25 


923919 


i-36 


811410 


4-6? 


188590 


4 


57 


54391 


839i5 


735525 


3-25 


923837 


i-36 


811687 


4-6? 


f883i 3 


3 


58 


544 1 5 


83899 


735719 


3-24 


923755 ; 


i-3 7 


81 1964 


4-61 


i88o36 


2 


59 


54440 


83883 


735914 


3-24 


923673 


1 .,37 


812241 


4-61 


187759 


1 


IS 


54464 


83867 


736109 


3-24 


92359-I 


r.3 7 


812017 


4-6t 


187483 


° j 


C~ 


N. cos. 2f. sine. 


L. cos. 


D. 1" 


Ia sine. 


L. cat. 


D.l" 


L. tang. 


, 


i 




5?° 









SINES AND TANGENTS. 



-33< 



51 



Rad. = 1. 


Logarithms. — Rauius = 10 10 . 


' 


N.sine.[ N. cos. 


L. sine. 


D. V 


L. cos. D 


1" 


L. tang. 


D. 1" 


L. cot. 







54464 ! 83867 


9-736109 
7363o3 


3-24 


9-923591 I 


37 


9.812517 


4-6i 


10-187483 


60 


1 


'54488:8385i 


3-24 


923509 1 


37 


812794 


4-6i 


187206 


59 


2 


; 545i3 |83835 


736498 


3-24 


923427 1 


37 


813070 


4-6i 


186930 


58 


3 


154537 838i 9 


736692 


3-23 


923345 1 


37 


8i3347 


4-6o 


186653 


57 


4 


5456i 838o4 


736886 


3-23 


923263 1 


37 


8i3623 


4-6o 


186377 


56 j 


5 


54586 ; 83788 


737080 


3-23 


9 23i8i 1 


37 


813899 
8i4n5 


4-6o 


186101 


55 | 


6 


| 54610 , 83772 


737274 


3-23 


923098 1 


37 


4-6o 


185825 


54 j 


7 


: 54635 j 83756 


737467 


3-23 


923016 1 


37 


8i4452 


4-6o 


185548 


53 


8 


5465 9 ! 83740 


737661 


3-22 


922933 1 


37 


814728 


4-6o 


185272 


52 


9 


54683 ! 83724 


7 3 7 855 


3-22 


92285i 1 


37 


816004 


4-6o 


184996 


51 


10 
11 


! 54708 ; 83708 
1 54732 836 9 2 


738048 
9-738241 


3-22 


922768 1 


38 
38 


815279 


4-6o 


184721 


50 
49 


3-22 


9-922686 1 


9-81 5555 


4.59 


10-184445 


12 


i 54756 i 83676 


738434 


3-22 


922603 1 


38 


8i583i 


4-5g 


1 841 69 
183893 


4S 


13 


S 54781 8366o 


738627 


3-21 


922520 1 


38 


816107 


4-5 9 


47 


14 ' 548o5 j 83645 


738820 


3-21 


922438 1 


38 


8i6382 


4-5 9 


i836i8 


46 


15 1; 54829 !8362 9 


739013 


3-21 


922355 1 


38 


8 1 6658 


4-59 


183342 


45 


16 j: 54854 !836i3 


739206 


3-21 


922272 1 


38 


8i6 9 33 


4- 5 9 


183067 


44 


17 54878183597 


739398 


3-21 


922189 1 


38 


817209 


4-5 9 


182791 


43 


18 54902 |8358i 


739590 


3-20 


922106 1 


38 


817484 


4-59 


182516 


42 


19 54927 183565 


73 97 83 


3-20 


922023 1 


38 


817759 
8i8o35 


4-5 9 


182241 


41 


20 i 54g5i 183549 

21 1 54975 i 83533 


739975 


3-20 
3-20 


921940 1 


38 


4-58 


181965 


40 


9-740167 


9-921857 1 


39 


9 -8i83io 


4-58 


10-181690 


3y 


22 I 54999 |835i7 


740359 


3-20 


921774 1 


3 9 


8i8585 


4-58 


i8i4i5 


38 


23 !' 55o24 !835oi 


74o55o 


3-19 


921691 1 


39 


818860 


4-58 


1 8 1 1 40 


37 


24 ! 55o48 : 83485 


740742 


3-ig 


921607 1 


3 9 


819135 


4-58 


i8o865 


36 


25 j 55072 83469 


740934 


3-19 


921524 1 


3 9 


819410 


4-58 


180590 


35 


26 1 55097 83453 


741 125 


3-19 


921441 1 


3 9 


819684 


4-58 


i8o3i6 


34 


27 | 55i2i 183437 


74i3i6 


3-19 


921357 1 


3 9 


819959 


4-58 


1 8004 1 


33 


28 |55i45 83421 


74i5o8 


3- 18 


921274 1 


39 


820234 


4-58 


179766 


32 


29 j! 55169 834o5 


741699 


3-1-8 


92 1 1 90 1 


3 9 


82o5o8 


4-57 


179492 


31 


30 j| 55194 83389 

31 ! 552i8 : 83373 


741889 


3-i8 
3-i8 


921107 1 


39 
3 9 


820783 


4-57 


179217 


30 


9-742080 


9-921023 1 


9-821057 


4-57 


io- 178943 


29 


32 55242 83356 


742271 


3- 18 


920939 1 


40 


821332 


4-57 


178668 


28 


33 i : 55266 ! 83340 


742462 


3-i 7 


920856 1 


40 


821606 


4-57 


i 7 83 9 4 


27 I 


34 1 55291 83324 


742652 


3-17 


920772 1 


40 


821880 


4-57 


178120 


26 


35 |!553i5 !833o8 


742842 


3-17 


920688 1 


40 


822154 


4-57 


177846 


25 J 


36 |: 55339 83292 


743o33 


3-i 7 


920604 1 


40 


822429 
822708 


4-57 


177571 


24 I 


j 37 1 55363 83276 


743223 


3-17 


920520 1 


4o 


4-57 


177297 


23 § 


1 3S |j 55388 ; 8326o 


7434i3 


3-i6 


920436 1 


40 


822977 


4-56 


177023 


22 J 


39 1 55412 


83244 


743602 


3-i6 


92o352 i 


40 


82325o 


4-56 


176730 


21 | 


40 !j 55436 
S 41 ij 05460 


83?28 

83212 


743792 


3- 16 
3-i6~ 


920268 1 
9-920184 i 


40 
40 


823524 


4-56 


176476 


20 I 


9-743982 


9-823 79 8 


4-56 


io- 176202 


19 I 


42 |j 55484 


83i95 


744i 7 1 


3-i6 


920099 1 


40 


824072 


4-56 


175028 


18 I 


43 555o9 


83179 


74436i 


3-i5 


920015 1 


40 


824345 


4-56 


i756o5 17 | 


44 |i 55533 


83i63 


74455o 


3 - 1 5 


919931 1 


41 


824619 


4-56 


i7538i 


16 


j 45 j 55557 


83 147 


744739 


3 - 1 5 


919846 1 


4i 


8248q3 


4-56 


175107 


15 I 


S 46 5558 1 


83 1 3 1 


744928 


3-i5 


919762 1 


41 


825i66 


4-56 


174834 


14 f 


47 i 556o5 


83 1 1 5 


745ii7 


3-i5 


919677 1 


41 


825439 


4-55 


1 7456 1 


13 


48 1 5563o 


83098 


7453o6 


3-U 


919593 1 


4i 


825713 


4-55 


174287 


12 


49 j 55654 


83o82 


745494 


3-U 


919508 1 


41 


825986 


4-55 


174014 


11 


50 1 55678 

51 ;i 55702 


83o66 
83o5o 


745683 
9-74587I 


3-14 
3.i4~ 


919424 1 


4i 
4* 


826259 


4-55 


i7374t 


10 
9 


9-919339 1 


9-826532 


4-55 


10-173468 


52 55726 


83o34 


746059 


3-i4 


919254 1 


41 


8268o5 


4-55 


173195 


8 


| 53 5575o 


83oi 7 


746248 


3-i3 


919169 1 


41 


827078 


4-55 


172922 


7 


54 ! 55775 


83ooi 


746436 


3-i3 


919085 1 


41 


827351 


4-55 


172649 


6 


55 | 55799 


82985 


746624 


3-i3 


919000 1 


41 


827624 


4-55 


172376 


5 


56 1 55823 


82969 


746812 


3-i3 


918915 1 


42 


827897 


4-54 


172103 


4 


57 : 55847 


82953 


746999 
747187 


3-i3 


9 i883o 1 


42 


828170 


4-54 


171830 


3 


58 !J 55871 


82 9 36 


3-12 


918745 1 


42 


828442 


4-54 


I7i558 


2 


59 J 55895 


82920 


1 747374 


3-12 


918659 1 


42 


828715 


4-54 


171285 


1 


60 | 55919 


82904 


1 747562 


3-12 


918574 li 


42 


828987 


4-54 


171013 





j N. cos. 


N. sine. 


I L. cos. 


D. 1" 


L. sine. 




L. cot. 


D. \" 


L. tang. 


' 


56° 












19* 













52 



SINES AND TANGENTS. 34 ( 



Rad.=1. 




LOUARITHMS.- 


-Radius = 10'°. 




— 


N. sine. 


N. cos. 


L. sine. 


D. 1" 


L. cos. ] 


).l" 


L. tang. 


D. 1" 


L. cot. 







55919 


82904 


9.747562 


3 


12 


9-918574 I 


•42 


9-828987 


4 


54 


10-171013 


60 


1 


55943 


82887 


747749 


3 


12 


918489 1 


•42 


829260 


4 


54 


170740 


59 


2 


55 9 68 


82871 


747936 


3 


12 


918404 1 


•42 


829532 


4 


54 


170468 


58 





55992 


82855 


748123 


3 


II 


9i83i8 1 


•42 


829805 


4 


54 


17019D 


57 


4 


56oi6 


82839 


7483io 


3 


II 


918233 1 


•42 


830077 


4 


54 


169923 


56 


5 


56o4o 


82822 


748497 


3 


II 


918147 1 


•42 


83o349 


4 


53 


169651 


55 


6 


56o64 


82806 


748683 


3 


n 


918062 1 


•42 


83o62i 


4 


53 


169379 


54 


7 


56o88 


82790 


748870 


3 


11 


917976 1 
917891 1 


•43 


830893 


4 


53 


169107 


53 


8 


56ii2 


82773 


749056 


3 


10 


•43 


83n65 


4 


53 


168835 


52 


9 


56i36 


82757 


749243 


3 


10 


917805 1 


•43 


83 1 437 


4 


53 


168563 


5L 


10 


56 1 60 

56i84 


82741 
82724" 


749429 

9-749615 


3 


10 


917719 1 


•43 


831709 


4 


53 


168291 


50 


11 


3 


10 


9-917634 1 


•43 


"9T83198T 


4 


53 


10-168019 


49 


^ i 


562o8 


82708 


749801 


3 


10 


917548 1 


-43 


832253 


4 


53 


167747 


48 


1 ; > 


56232 


82692 


749987 


3 


09 


917462 1 


•43 


832525 


4 


53 


167475 


47 


14 t 


56256 


82675 


750172 


3 


09 


917376 1 


•43 


832796 


4 


53 


167204 


46 


15 | 


5628o 


82659 


75o358 


3 


09 


917290 1 


•43 


833o68 


4 


52 


166932 


45 


16 1 


563o5 


82643 


75o543 


3 


09 


917204 1 


•43 


833339 


4 


52 


1 6666 1 


44 


17 i 


56329 


82626 


750729 


3 


3 


917118 1 


•44 


8336i 1 


4 


52 


i6638o 


43 


18 


56353 


82610 


750914 


3 


917032 1 


•44 


833882 


4 


52 


166118 


42 


19 | 


56377 


825 9 3 


751099 


3 


08 


916946 1 


•44 


834i 54 


4 


52 


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41 


20 


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82577 


751284 


3 


08 


916859 1 


•44 
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834425 
"9^834696" 


4 

4 


52 

52" 


165575 
io- i653o4 


40 
39 


21 


56^25 


82~56T 


9-751469 


3 


08" 


9-916773 


1 "^ 


56449 


82544 


75i654 


3 


08 


916687 


•44 


834967 


4 


52 


i65o33 


38 


\ 28 ; 


56473 


82528 


75i83g 


3 


08 


9 1 6600 1 


•44 


835238 


4 


52 


164762 


37 


I 24 ! 


56497 


82D1 1 


752023 


3 


07 


916514 1 


•44 


835509 


4 


52 


1 6449 1 


36 


25 ' 


56521 


82495 


752208 


3 


07 


916427 i 


•44 


835 7 8o 


4 


5i 


164220 


35 


It 1 


56545 


82478 


752392 


3 


07 


916341 1 


•44 


836o5i 


4 


5i 


163949 


34 


27 


56569 


82462 


752576 


3 


07 


916254 1 


•44 


836322 


4 


5i 


163678 


33 


28 ! 


565 9 3 


82446 


752760 


3 


07 


916167 1 


•45 


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4 


5 1 


163407 


32 


29 ! 


D6617 


82429 


752944 


3 


06 


916081 1 


•45 


836864 


4 


5 1 


i63i36 


31 


j SO i 


5664i 
56665 


82413 
"8~23q6" 


753i28 
9-7533i2 


3 
3 


06 
"06 


9 l5 994 1 
9-915907 j 


•45 
•45 


837i34 


4 


5i 


162866 


30 


9- 8374o5 


4 


5i 


10-162595 


29 


32 


56689 


8238o 


753495 


3 


06 


916820 


•45 


837675 


4 


5i 


162325 


28 


33 


56713 


82363 


753679 


3 


06 


9i5733 


•45 


837946 


4 


5i 


162054 


27 


34 


56736 


82347 


753862 


3 


o5 


915646 


•45 


838216 


4 


5 1 


161784 


26 


35 


56760 


82330 


754046 


3 


o5 


915559 


•45 


838487 


4 


5o 


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25 


36 


56784 


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3 


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4 


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24 


37 


568o8 


82297 


754412 


3 


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9i5385 


•45 


839027 


4 


5o 


160973 


23 


38 


56832 


82281 


754095 


3 


o5 


915297 


•45 


839297 


4 


5o 


160703 


22 


39 


56856 


82264 


754778 


3 


04 


9i52io 


•45 


83 9 568 


4 


5o 


160432 


21 


40 


5688o 


82248 
8223 1 


754960 
9-755i43 


3 


04 


9i5i23 


-46 
•46 


83 9 838 
9-840108 


4 
4 


5o 
"5o~ 


160162 

io- 159892 


20 

19 


41 


56904 


3 


04 


9-9i5o35 


42 


56928 


82214 


755326 


3 


04 


914948 


• 46 


840378 


4 


5o 


159622 


IS 


43 


56952 


82198 


7555o8 


3 


04 


914860 


-46 


840647 


4 


5o 


i59353 


17 


44 


56976 


82181 


755690 


3 


04 


914773 


-46 


840917 


4 


40 


i5qo83 


16 


45 


57000 


82i65 


755872 


3 


o3 


914685 


-46 


841187 


4 


49 


i !$.8i 3 


15 


4fi 


57024 


82148 


756o54 


3 


o3 


914598 


-46 


841457 


4 


49 


1 58543 


14 


17 


57047 


82132 


756236 


3 


o3 


9i45io 


•46 


841726 


4 


49 


158274 


13 i 


48 


57071 


82ii5 


756418 


3 


o3 


9U422 


•46 


841996 


4 


49 


1 58oo4 


12 


49 


57095 


82098 


756600 


3 


o3 


914334 


• 46 


842266 


4 


49 


i5 77 34 


11 


50 


57119 


82082 


756782 


3 


•02 


914246 


•47 


842535 


4 


49 


1 57465 


10 


51 


57U3 


82065 


9-756963 


3 


02 


9-9i4i58 


•47 


9-842805 


4 


49 


io- 1D7195 


9 


52 


57167 


82048 


757U4 


3 


02 


914070 


•47 


843074 


4 


49 


156926 


8 


53 


5 7 igi 


82032 


757326 


3 


02 


913982 


•47 


843343 


4 


49 


166607 


7 


54 


57215 


82015 


757507 


3 


02 


9 i38 9 4 


•47 


843612 


4 


49 


156388 


6 


55 


57238 


81999 
81982 


757688 


3 


01 


9l38o6 


•47 


843882 


4 


48 


i56n8 


5 


56 


57262 


757869 


3 


01 


913718 


•47 


8441 5i 


4 


48 


1 55849 


4 


57 


57286 


81963 


758o5o 


3 


01 


9i363o 


•47 


84.I.I20 


4 


48 


1 5558o 


S 


58 


57310 


81949 


75823o 


3 


• 01 


9i354i 


•47 


844689 

844958 


4 


48 


1 553 11 


2 


59 


5 7 334 


8i 9 32 


75841 1 


3 


•01 


9i3453 


•47 


4 


48 


i55o42 


1 


00 


57358 
N. cos. 


8igi5 

N. sine 


758591 


3-oi 


913365 


•47 


845227 


4-48 


154773 





L. cos. 


D. 1" 


L. sine. 




L. cot. 


D. 1" 


L. tang. 


' 


55° 



SINES AND TANGENTS. 35 c 



53 



Rad. = 1. 


j 




Logarithms.— 


-Radius = 10 10 . 




' 


N. sine. 


N. eos. 


! L. sine. 


D. 1" 


L. cos. 


D.l" 


L. tang. 


D.l" 


L. cot 







j 57358 


8i 9 i5 


9-758591 


3-oi 


9-9i3365 


i-47 


9-845227 


4-48 


10- 154773 


60 


1 1 07381 


8! 899 


758772 


3-oo 


913276 


i-47 


845496 


4-48 


1 545o4 


59 


2 j 574o5 


81882 


758952 


3- 00 


913187 


1-48 


845764 


4-48 


154236 


58 


8 j 57429 


81 865 


759132 


3-oo 


9 1 3099 


1-4.8 


846o33 


4-48 


15J967 
1 536 9 8 


m 


4 ;; 57453 


85848 


739312 


3-oo 


9i3oio 


1-48 


8463 02 


4-48 


56 


i j 157477 


8i832 


759492 


3-co 


912922 


1-48 


846570 


4-47 


i5343o 


55 


6 ; 37301 


8i8i5 


759672 


2-99 


912833 


1-48 


846839 


4-47 


z53i6i 


54 


7 


57524 


81798 


759852 


2-99 


912744 


1.48 


847107 


4-47 


152893 


53 


8 


57548 


81782 


76003 1 


2-99 


912655 


1-48 


847376 


4-47 


152624 


52 


i> 


57572 


81765 


7602 1 1 


2-99 


912366 


1-48 


847644 


4-47 


i52356 


51 


10 


57596 


81748 


760390 


2-99 


912477 


1-48 


847913 


4-47 


152087 


50 


11 


57619 


81731 


[ 9-760369 


2-98 


9-912388 


1^48 


9-848181 


4-47 


io-i5i8i9 


49 


12 


57643 


8i 7 i4 


760748 


2-98 


912299 


1-49 


848449 


4-47 


i5i55i 


48 


13 


57667 


81698 


760927 


2-98 


912210 


1-49 


848717 


4-47 


i5i283 


47 


14 


57691 


81681 


761106 


2-98 


912121 


1-49 


848986 


4-47 


i5ioi4 


46 


15 


J5 77J 5 


81664 


761285 


2-98 


912031 


1.49 


849234 


4-47 


1 50746 


45 


J 16 


! 57738 


81647 


761464 


2-98 


91 1942 


1-49 


849522 


4-47 


130478 


44 


1 17 


157762 


8i63i 


761642 


2 -97 ; 


911853 


1-49 


84979° 


4-46 


l502I0 


43 


I 18 II 5 77 86 


81614 


761821 


2-97 


91 1763 


1-49 


85oo58 


4-46 


149942 


42 


j 19 5 7 8io 


8i5 97 


761999 


2.97 


911674 


1-49 


85o325 


4.46 


149675 


41 


I 20 


|5 7 833 


8i58o 


762177 


2-97 


91 1 384 


1-49 


8 5o593 


4.46 


149407 


40 


21 


! 57857 


8 1 563 


9-762306 


2-97 


9-911495 


1-4-9 


9-85o86i 


4.46 


io-i49i3g 


39 


22 


5 7 88i 


8 1 546 


762534 


2-96 


911405 


1-49 


85i 129 


4-46 


1 4887 1 


38 


23 


1 57904 


8i53o 


762712 


2-96 


91 13 1 5 


i«5o 


85 1 396 


4-46 


148604 


37 


24 


57928 


8i5i3 


762889 


2-96 


911226 


i-5o 


85 1 664 


4-46 


U8336 


36 


25 


57952 


81496 


763067 


2-96 


911 i36 


i-5o 


85i93i 


4.46 


148069 


35 


26 


57976 


81479 


763245 


2-96 


9 1 1 046 


i-5o 


852199 


4-46 


147801 


34 


27 


57999 


81462 


763422 


2-96 


910956 


i -5o 


852466 


4.46 


U7534 


33 


28 


58023 


8.445 


7636oo 


2-95 


910866 


i-5o 


852733 


4-45 


147267 


32 


29 


98047 


81428 


763777 


2-95 


910776 


i-5o 


853ooi 


4-45 


146999 


31 


30 
31 


58070 


81412 


763954 


2-95 
2-95 


910686 


i-5o 


853268 


4-45 


146732 


30 


58og4 


81395 


9-764131 


9-910596 


i-5o 


9-853535 


4-45 


10-146465 


29 


32 


58n8 


8i3 7 8 


764308 


2-93 


9io5o6 


1 -5o 


8538o2 


4-45 


146198 


28 


33 


58i4i 


8i36i 


764485 


2-94 


gio4i5 


i-5o 


854069 


4-45 


1 4593 1 


27 


34 


58i65 


8 1 344 


764662 


2-94 


9io325 


i-5i 


854336 


4-45 


145664 


26 


35 


I 58189 1 81327 


7 64838 


2-94 


910235 


i-5i 


8546o3 


4-45 


145397 


25 


36 


j 582 1 2 | 8i3io 


765oi5 


2-94 


1 1 44 


i-5i 


854870 


4-45 


!45i3o 


24 


37 


58236 8i2 9 3 


765191 


2-94 


910034 


i-5i 


855i37 


4-45 


144863 


23 


38 


138260 181276 


765367 


2-94 


909963 
909873 


1 -5i 


855404 


4-45 


144596 


22 


39 


58283! 8i25 9 


765544 


2-93 


1 -5i 


855671 


4.44 


144329 


21 


40 
41 


583o7 | 81242 


765720 
9-765896 


2-93 


909782 


i-5i 
i-5i 


855938 


4-44 


144062 


20 


5833o 1 81225 


2-93 


9-909691 


9-856204 


4-44 


10-143796 


19 


42 


58354 


81208 


766072 


2-93 


909601 


[-5! 


856471 


4-44 


143529 


18 


43 


58378 


81191 


766247 


2- 9 3 


909510 


i-5i 


856737 


4-44 


143263 


17 


44 


584oi 


81x74 


766423 


2- 9 3 


909419 


i-5i 


857004 


4-44 


142996 


16 


45 


58425 


8il5f 


766598 


2-92 


909328 


1-52 


857270 


4-44 


142730 


15 


46 


58449 


81140 


766774 


2-92 


909237 


1-52 


85 7 53 7 


4.44 


U2463 


14 


47 


5S472 


8ii23 


766949 


2-92 


909146 


1-52 


857803 


4.44 


142197 13 
I4i93x 12 


48 


58496 


81106 


767124 


2-92 


909055 


1-52 


858069 


4-44 


49 


585i9 


81089 


767300 


2-92 


908964 


1-52 


858336 


4.44 


141664 11 


50 
51 


58543 
: 58567 


81072 


767475 


2-91 


908873 


I -52 
1-52 


8586o2 


4.43 


141398 


10 


8io55 


9-767649 


2-91 


9-908781 


9-858868 


4-43 


io-i4ii32 


9 


52 


58590 


8io38 


767824 


2-91 


908690 


1-52 


859134 


4-43 


140866 


8 


53 


586i4 


81021 


767999 


2.91 


908599 


£-52 


859400 


4-43 


140600 


7 


54 


58637 


81004 


768173 


2-91 


908507 


1-52 


85 9 666 


4-43 


i4o334 


6 


55 


1 5866i 


80987 


768348 


2-90 


908416 


1 -53 


859932 


4-43 


140068 


5 


56 


i 58684 


80970 


768322 


2-90 


908324 


i-53 


860198 


4-43 


139802 


4 


57 


58708 


8o 9 53 


768697 


2-90 


908233 


i-53 


860464 


4-43 


i3 9 536 


3 


58 


58731 


8o 9 36 


768871 


2-90 


908141 


1-53 


860730 


4.43 


139270 


2 


59 


58755 


80919 


769043 


2-90 


908049 


1-53 


860995 


4-43 


139005 
i38 7 3 9 


1 


60 


58779 


80902 


769219 


2-90 


907958 


1-53 


861261 


4-43 







N. cos. 


N.sine. 


L. cos. 


D.l" 


L. sine. 




L. cot. 


D.l" 


L. tang. 


' 


54° 



54 



SINES AND TANGENTS. — 36°„ 



Rad.^1. 




LOGAKITHMS.— 


-Radius = 10 ,n . 







N. sine. 


N. cos. 


L. sine. 


D 


1" 


L. cos. D 


1" 


L. tang. 


Dl." 


L. cot. 




, 58779 


80902 


9-769219 


2 


90 


9-907958 I 


53 


9-861261 


4-43 


10-138739 


60 


1 


! 58802 


8o885 


769393 


2 


89 


907866 1 


53 


861527 


4-43 


138473 


59 


2 


! 58826 


80867 


769566 


2 


89 


907774 1 


53 


861792 
862058 


4-42 


138208 


58 


3 


58849 


8o85o 


769740 


2 


89 


907682 l 


53 


4-42 


J 3794 2 


57 


4 


! 58873 


8o833 


769913 


2 


8 o 9 


907590 I 


53 


862323 


4-42 


137677 


56 


5 


! 58896 


80816 


770087 


2 


89 


907498 1 


53 


862589 


4-42 


I374I1 


55 


6 


! 58920 


80799 


770260 


2 


88 


907406 1 


53 


862854 


4-42 


l37U6 


54 


7 


; 58 9 43 


80782 


770433 


2 


88 


907314 I 


54 


863ii9 


4-42 


1 36881 


53 | 


8 


| 58967 


80765 


770606 


2 


88 


907222 I 


54 


863385 


4-42 


i366i5 


52 I 


9 


58990 


80748 


770779 


2 


88 


907129 l 


54 


863'65o 


4-42 


i3635o 


51 


10 
11 


D9014 

09037 


80730 
8ot~iT 


770902 


2 


88 


907037 1 


04 


8639i5 
9 '864180 


4-42 


i36o85 


50 


9-771125 


2 


88 


9-906945 I 


54 


4-42 


io- j358?o 


49 


12 


09061 


80696 


77 1 298 


2 


87 


906852 I 


04 


864445 


4.42 


i35555 


4S 


13 


59084 


80679 


771470 


2 


87 


906760 I 


54 


864710 


4-42 


135290 


47 


14 


59108 


80662 


771643 


2 


87 


906667 I 


54 


864975 


4-4i 


i35o25 


46 


15 


5913 1 


80644 


77181D 


2 


87 


906575 I 


04 


865240 


4-4i 


134760 


45 


16 


091 54 


80627 


771987 


2 


87 


906482 1 


04 


8655o5 


4-4i 


134495 


44 


17 


59.78 


80610 


772109 


2 


87 


906389 I 


00 


865770 


4-4i 


i3423o 


43 


18 


5920c 


8o5 9 3 


77233i 


2 


86 


906296 1 


55 


866o35 


4'4i 


133965 


42 


19 


D9225 


80576 


7725o3 


2 


86 


906204 I 


55 


8663oo 


4-4i 


133700 


41 


20 


59248 


8o558 


772675 


2 


86 


9061 1 1 I 


00 


866564 


4-41 


133436 


40 


21 


59272 


8o54i 


9.772847 
773018 


2 


86 


9-906018 1 


55 


9-866829 


4-4i 


IO- I 33l7I 


~3y~ 


22 


D92()5 


8o524 


2 


86 


9o5g25 I 


55 


867094 


4-4i 


132906 


38 


23 


5 9 3i8 


80307 


773190 


2 


86 


905832 1 


55 


867358 


4-41 


132642 


37 


24 


59342 


80489 


77 336i 


2 


85 


905739 1 


55 


867623 


4-4* 


132377 


36 


25 


59365 


8o4"2 


773533 


2 


85 


905640 1 • 


55 


867887 


4-4i 


i32ii3 


35 


26 


59389 


804.55 


773704 


2 


85 


905552 1 


55 


8681 52 


4-4o 


i3i848 


34 


27 


894 1 2 


8o438 


773875 


2 


85 


905459 1 


55 


868416 


A-Ao 


i3i584 


33 


28 


59436 


80420 


774046 


2 


85 


905366 1 


56 


868680 


4-4o 


i3i32o 


32 


29 


59459 


80403 


774217 


2 


85 


900272 1 


561 


868945 


4-4o 


i3io55 31 


30 
31 


O9482 


8o386 


774398 


2 
2 


84 
84" 


900179 1 


56! 
56 


869209 


4-4o 


130791 


30 


5 9 5o6 80368 


9-774558 


9'9o5o85 1 


9-869473 


4-4o 


io- i3o527 


yy 


32 


09529 8o35? 


774729 


2 


84 


904992 1 


56 


869737 


4-4o 


i3o263 


28 


33 


59552 8o334 


774899 


2 


84 


904898 1 


56 


87000 i 


4-4o 


1 29999 


27 


34 


59576 8o3i6 


. 775070 


2 


84 


904804 1 


56 


870265 


4-4o 


129735 


26 


35 


59099 80299 


775240 


2 


84 


9047 1 1 1 


56 


870529 


4-4o 


1 2947 1 


25 


36 


59622 ; 80282 


775410 


2 


83 


9046 1 7 1 


06 


870793 


4-4o 


129207 


24 


37 


59646 j 80264 


77558o 


2 


83 


904523 1 


56 


871057 


4-4o 


128943 


23 


38 


09669 I 80247 


770750 


2 


83 


904429 1 


57 


8 7 i32i 


4.40 


128679 


22 


39 


59693 


8o23o 


775920 


2 


83 


904330 1 


57 


8 7 i585 


4-4o 


1 284 1 5 


21 


40 

IT 


,59716 


80212 


776090 


2 


83 


904241 1 


57 


871849 


4-3 9 


i28i5i 


20 


; 59739 80 195 


9-776209 


2 


83 


9-904147 ' 


5 7 


9-8721 12 


4-3 9 


10-127888 


19 


42 


! 59763 80178 


776429 


2 


82 


9o4o53 1 


57 


872376 


4-3 9 


127624 


18 


43 


1 59786 80160 


776098 


2 


82 


903959 1 


57 


872640 


4-3 9 


127360 


17 


44 


5 9 8oq 80143 


776768 


2 


82 


9o3864 1 


5 7 


872903 


4-39 


1 27097 


16 


45 


59832 ! 80125 


776937 


2 


82 


903770 1 


57 


8 7 3i6 7 


4-3 9 


1 26833 


15 


1 46 


5 9 856: 80108 


777106 


2 


82 


90.3676 1 


57 


873430 


4-3 9 


126570 


14 


47 


59879 80091 


77727O 


2 


81 


9o358i 1 


5 7 


873604 


4-3 9 


rz63o6 


18 


48 


59902 : 80073 


777444 


2 


81 


903487 1 


^7 


8 7 3 9 5 7 


4-3 9 


126043 


12 


3 49 


59926 8oo56 


77 7 6i3 


2 


81 


9o33g2 1 


58 


874220 


4-3 9 


125780 


11 


llo 

i 5i 


59949 ! 8oo38 


777781 


2 


81 


90329^ 1 


58 


874484 


4-39 


1255 1 6 


1Q ■ 


59972 : 80021 


| 9-7779™ 


2 


81 


9-903203 1 


58 


9-874747 


4-3 9 


10- 12.0253 


9 . I 


1 52 


59990 : 8ooo3 


] 778119 


2 


81 


903 108 ! 


58. 


875010 


4-3 9 


124990 


53 


60019 79986 


778287 


2 


80 


9o3oi4 1 


5* 


875273 


4-38 


124727 ! 1 


54 


60042 7996S 


778455 


2 


80 


902919 1 


OS 


875536 


4-38 


124464 ' :; 


55 


60060 79901 


77^624 


2 


80 


902824 1 


58 


870800 


4-38 


124200 5 ( 


55 


60089 i 799^4 


778792 


2 


80 


902729 1 


58 


876063 


4-38 


123937 


4 


57 


601 12 79916 


778960 


2 


80 


992634 1 


58 


876326 


4-38 


123674 


3 


58 


6oi35 79899 


779128 


2 


80 


902539 1 


5g 


876589 


4-38 


123411 


2 


59 


! 60 1 58| 79881 


779290 


2 


79 


902444 1 


00 


876851 


4-38 


123149 


1 


60 


1 60182 179864 


779463 


2-79 
D. 1 


902349 1 


5g 


877114 


4-38 


722886 





B 


1 


L. sine 1 




L. cot. ! d. r 


L. tang. 




1 




53° 








i 



SINES AND TANGENTS. — 37°. 



55 



Rad.=1. 






Logarithms.— 


-Radius = 10 10 . 




' 


N. sine. 


N. cos. 


L. sine. 


D.l" 


L. cos. 


D.l" 


L. tang. 


D.l" 


L. cot. 







60182 


79864 


9-779463 


2 


79 


9-902349 


'& 


9-877114 


4-38 


10-122886 


60 


1 


6o2o5 


79846 


779631 


2 


79 


902253 


•5 9 


877377 


4-38 


122623 


59 


2 


60228 


79829 


779798 


2 


79 


902158 


I 9 


877640 


4-38 


12 2360 


58 


8 


6o25i 


79811 


779966 


2 


79 


902063 


.5 9 


877903 


4-38 


122097 
121835 


57 


4 


60274 


79793 


78oi33 


2 


79 


901967 


-5 9 


878165 


4-38 


56 


5 


60298 


79776 


780300 


2 


78 


901872 


-5 9 


878428 


4-38 


121D72 


55 


6 


6o32i 


7 97 58 


780467 


2 


78 


901776 


• 5 9 


878691 


4-38 


121309 


54 


7 


I 6o344 


79741 


780634 


2 


78 


901681 


•5 9 


878953 


4-37 


121047 


53 


8 


j 60367 


79723 


780801 


2 


78 


90i585 


•5 9 


879216 


4-3 7 


120784 


52 


9 


60390 


79706 


780968 


2 


78 


901 490 


•5 9 


879478 


4-37 


120522 


51 


10 
11 


j 60414 


79688 


781134 


2 


78 


901394 


•60 


879741 


4-3 7 


120259 50 


60437 


79671 


9«78i3oi 


2 


77 


9-90^298 


•6o 


9-88ooo3 


4-37 


10-119997 1 49 


12 


60460 


79653 


781468 


2 


77 


905202 


•6o 


880265 


4-37 


119735 


48 


13 


6o483 


79635 


781634 


2 


77 


90 8 106 


•60 


88o528 


4-3 7 


II9472 


47 


14 


6o5o6 


79618 


781800 


2 


77 


9OIOIO 


-6o 


880790 


4-37 


119210 


46 


15 


60629 


79600 


781966 


2 


77 


9OO9I4 


•60 


88io52 


4-3 7 


1 1 8948 


45 


16 


j 6o553 


79583 


782132 


2 


77 


9O0818 


•60 


8Si3i4 


4-3 7 


1 1 8686. 


44 


17 


60576 


79565 


782298 


2 


76 


9OO722 


•6o 


881576 


4-37 


1 18424 


43 


18 


j 60D99 


79547 


782464 


2 


76 


900626 


■60 


88i83 9 


4-3 7 


118161 


42 


19 


I 60622 


7953o 


782630 


2 


76 


900529 


-6o 


882101 


4-37 


117899 


41 


20 


6o645 


7 9 5i2 


782796 


2 


76 


900433 


•61 


882363 


4-36 


u 7 63 7 


40 


21 


60668 


79494 


9-782961 


2 


76 


9-900337 1 


■61 


9-882625 


4-36 


io- 117375 


39 I 


22 


60691 


79477 


783.27 


2 


76 


900240 


•61 


882887 


4-36 


1 171 13 


SS I 


23 


60714 


79459 


783292 


2 


75 


900144 ! 


•61 


883i48 


4-36 


n6852 


37 I 


24 


6o 7 38 


79441 


783458 


2 


7 5 


900047 1 


•6i 


8834io 


4-36 


116590 


36 I 


25 1160761 


79424 


783623 


2 


7 5 


89995 1 1 


• 61 


8836 7 2 


4-36 


116328 


35 I 


26 1 60784 


79406 


783788 


2 


75 


899854 1 


•61 


883 9 34 


4-36 


116066 


34 I 


2tf 60807 


79388 


783 9 53 


2 


75 


899757 


-6i 


884196 


4-36 


n58o4 


33 I 


28 6o83o 


79371 


7841 18 


2 


75 


899660 1 


•61 


884457 


4-36 


ii5543 


32 | 


29 !6o853 


79353 


784282 


2 


74 


899564 


• 6i 


884719 


4-36 


i.l52.8i 


31 i 


30 


60876 


79335 


784447 


2 


74 


899467 1 


• 62 
-62 


884980 


4-36 


ll5020 


30 j 
29 I 


31 


60899 


79318 


9-784612 


2 


74 


9-899370, 1 


9-885242 


4-36 


IO- H4758 


32 


60922 


79300 


784776 


2 


74 


899273 1 


• 62 


8855o3 


4-36 


1 14497 


28 I 


83 


60945 


79282 


784941 


2 


74 


899176 1 


.62 


885765 


4-36 


1 14235 


27 I 


34 


60968 


79 26 4 


785io5 


2 


74 


899078 I 


• 62 


886026 


4-36 


1 13974 


26 j 


35 


1 60991 


79 2 47 


785269 


2 


73 


898981 I 


• 62 


886288 


4-36 


113712 


25 f 


36 


6ioi5 


79229 


785433 


2 


73 


898884 I 


•62 


886549 


4-35 


n345i 


24 j 


37 


6io38 


792 1 1 


785597 


2 


73 


898787 I 


• 62 


886810 


4-35 


1 1 3 1 90 


28 I 


38 


6 1 06 1 


79193 


785761 


2 


73 


898689 1 


-62 


887072 


4-35 


1 1 2928 


22 I 


39 


61084 


79176 


785925 


2 


73 


898592 1 


-62 


887333 


4-35 


1 1 2667 


21 [ 


40 


61107 


791 58 


786089 


2 


73 


898494 1 


• 63 


887594 


4-35 


1 1 2406 


20 I 


Tr 


61 i3o 


79'4o 


9-786252 


2 


72 


9 -898397 « 


^63 


9-887855 


4-35 


I0-H2I45 19 1 


42 


61 1 53 


79122 


786416 


2 


72 


898299 1 


•63 


888116 


4-35 


J 1 1884 IS f 


43 


61176 


79105 


786579 


2 


72 


898202 1 


•63 


888377 


4-35 


1 11623 17 I 


44 


61 199 


79087 


786742 


2 


72 


898104 ' 


• 63 


88863 9 


4-35 


iii36i 16 | 


45 


61222 


79069 


786906 


2 


72 


898006 1 


•63 


888900 


4-35 


1 11 100 15 4 


46 


61245 


79051 


787069 


2 


72 


897908 1 


•63 


889160 


4-35 


1 1 0840 1 14 | 


•i +> 


61268 


79o33 


787232 


2 


7'i 


897810 1 


• 63 


889421 


4-35 


110579 | 13 I 


48 


61291 


79016 


787395 


2 


7i 


897712 1 


•63 


889682 


4-35 


no3i8 12 1 


49 
50 


6i3i4 


78998 


787557 


2 


7i 


897614 1 


-63 


889943 


4-35 


110057 11 | 


6i33 7 


78980 


787720 


2 


7' 


8 97 5i6 1 


-63 


890204 


4-34 


109796 J 10 1 


. v-t 


6i36o 


78962 


9.787883 


2 


V 


9-897418 1 


"64 


9-890465 


4-34 


to- 109535 


9 1 


■ 52 


6i383 


78944 


788045 


2 


V 


897320 1 


•64 


890725 


4-34 


109275 


8 | 


" 


61406 


78926 


788208 


2 


V 


897222 1 


•64 


890986 


4-34 


1090U 


7 | 




61429 


78908 
78891 


788370 


2 


70 


897123 1 


•64 


891247 


4-34 


108753 


6 


55 


6i45i 


788532 


2 


70 


897025 1 


•64 


891507 


4-34 


lo8493 


5 


56 


61474 


78873 


788694 


2 


70 


896926 1 


-64 


891768 


4-34 


108232 


4 


57 


61497 


78855 


788856 


2 


70 


896828 1 


-64 


892028 


4-34 


107972 


3 


58 


6020 


78837 


789018 


2- 


70 


896729 1 


•64 


892289 


4-34 


107711 


2 ; 


59 


6i543 


78819 


789180 


2- 


70 


8 9 663 1 1 


•64 


892549 


4.34 


1 0745 1 


1 


«o_ 


6i566 


78801 


789342 


2-69 


896532 1 


•64 


892810 


4-34 


1 07 1 90 


: 




N. cos. 


N. sine. 


L. cos. 


D.l" 


L. sine. 




L. cot. 


D. 1" 


L. tang;. 


' 


E 










52° 









56 



SINES AND TANGENTS. — 38°. 



Rad. = 1. 




Logarithms.— 


-Radius =10 10 . 







N.sine. 


N. cos, 


L. sine. 


D. V 


L. cos. 


D.l" 


L. tang. 


D.l" 


L. cot. 




1-6*566 


78801 


9-789342 


2 


.69 


9-896532 


1-64 


! 9-892810 


4-34 


10-107190 
106930 


60 


9 i 


6i58 9 


78783 


789504 


2 


.69 


896433 


1-65 


893070 


4-34 


59 


I 2 


61612 


78765 


789665 


2 


.69 


896335 


1-65 


8g333l 


4-34 


106669 


58 


1 8 


l6r635 


78747 


789827 


2 


.69 


896236 


i-65 


893591 


4-34 


1 06409 


57 


I 4 


J6i658 


78729 


789988 


2 


.69 


896137 

8 9 6o38 


i-65 


8 9 385i 


4-34 


106149 


56 


I 5 


j 61681 


787 1 1 


790149 


2 


.69 


i-65 


8941 1 1 


4-34 


105889 


55 


1 6 


i 61704 


78694 


7903 10 


2 


•68 


895939 


i-65 


894371 


4-34 


105629 


54 


7 


61726 


78676 


j 790471 


2 


•68 


8 9 584o 


i-65 


894632 


4-33 


io5368 


53 


I 8 


j 61749 


78658 


790632 


2 


-68 


895741 


i-65 


894892 


4-33 


io5io8 


52 


1 9 


: 61772 


78640 


790793 


2 


• 68 


895641 


i.65 


895152 


4-33 


104848 


51 


I 10 
1 11 


i 61795 
618 1 8 


78622 
78604 


790954 


2 


• 68 


895542 


i-65 


895412 


4-33 


io4588 


50 


9.791115 


2 


• 68 


9-895443 


i-66 


9-895672 


4-33 


io-io4328 


49 


i 12 


,61841 


78586 


791275 


2 


•67 


895343 


i-66 


895932 


4-33 


104068 


48 


I 13 


61864 


78568 


791436 


2 


.67 


895244 


i-66 


896192 
896402 


4-33 


io38o8 


47 


I u 


61887 


7855o 


791596 


2 


•67 


895145 


i-66 


4-33 


io3548 


46 


1 15 


! 61909 


78532 


791757 


2 


.67 


895045 


i-66 


896712 


4-33 


io3288 


45 


1 16 


61932 


785i4 


791917 


2 


67 


894945 


i-66 


896971 


4-33 


103029 


44 


17 


61955 


78496 


792077 


2 


67 


894846 


1-66 


897231 


4-33 


102769 


43 


I 18 


61978 


78478 


792237 


2 


66 


894746 


1-66 


897491 

897751 


4-33 


102509 


42 


I 19 


62001 


78460 


792397 


2 


■66 


894646 


1-66 


4-33 


102249 


41 


1 20 


62024 


78442 


792537 


2 


66 


894546 


1-66 


898010 


4-33 


101990 


40 


fW 


, 62046 


78424 


9-792716 


2 


66 


9-894446 


1.67 


9-898270 


4-33 


10-101730 


89 


1 22 


; 62069 


78405 


792876 


2 


66 


894346 


1.67 


8 9 853o 


4-33 


101470 


38 


23 


i 62092 


7 838 7 


793o35 


2 


66 


894246 


1.67 


898789 


4-33 


101211 


37 


1 24 


621 15 


78369 


793195 


2 


65 


894146 


1 -67 


899049 


4-32 


1 0095 1 


36 


| 25 


62i38 


7835i 


793354 


2 


65 


894046 


1-67 


8 99 3o8 


4-32 


100692 


85 


j 26 


1 62160 


78333 


7935i4 


2 


65 


893946 

893846 


1-67 


899568 


4-32 


100432 


34 


! 27 


: 62183 


7 83 1 5 


793673 


2 


65 


1.67 


899827 


4-32 


100173 


33 


28 


62206 


78297 


793832 


2 


65 


893745 


1.67 


900086 


4-32 


099914 


32 


29 

80 

! 31 


; 62229 


78279 


7 9 3 99 i 


2 


65 


8 9 3645 


1.67 


900346 


4-32 


099654 


31 


6225l 

1 62274 


78261 


7941 5o 


2 

2 


64 
64 


893544 


1-67 


900605 


4-32 


099395 


30 


78243 


9 -7943o8 


9-893444 


i-68 


9-900864 


4-32 


10 -099136 


29 


1 32 


62297 


78225 


794467 


2 


64 


893343 


i-68 


901124 


4-32 


098876 


28 


! 83 


62320 


78206 


794626 


2 


64 


8g3243 


i-68 


90 1 383 


4-32 


098617 


27 


34 


62342 


78188 


794784 


2 


64 


8 9 3i42 


i-68 


901642 


4-32 


o Q 8358 


26 


i 35 


62365 


78170 


794942 


2 


64 


893041 


1-68 


90 1 90 1 


4-32 


098099 


25 


36 


62388 


78152 


795101 


2 


64 


892940 


1-68 


902160 


4-32 


097840 


24 


87 


62411 


7 8i34 


795259 


2 


63 


892839 


1.68 


902419 


4-32 


097581 


23 


38 


62433 


781 16 


793417 


2 


63 


892739 


1-68 


902679 


4-32 


097321 


22 


30 


62456 


78098 


795575 


2 


63 


8 9 2638 


1-68 


902938 


4-32 


097062 


21 


40 


62479 

62302 


78079 
7806? 


795733 


2 
2 


63 

~63~ 


892536 
9-892435 


1-68 
i-6 9 


903197 


4-3i 
~4737~ 


096803 


20 


9-795891 


9-9o3455 


10-096545 


19 


42 


62524 


78043 


796049 


2 


63 


892334 


1.69 


903714 


4-3i 


096286 


18 I 


143 


62547 


78025 


796206 


2 


63 


892233 


1.69 


903973 


4-3i 


096027 


17 


44 


62570 


78007 


796364 


2 


62 


892132 


1.69 


904232 


4-3i 


095768 


16 


45 


62592 


77988 


796521 


2 


62 


892030 


1.69 


904491 
904750 


4-3i 


095509 


15 


46 


626l5 


77970 


796679 


2 


62 


891929 


i-6 9 


4-3i 


095250 


14 


|47 


62638 


7 79 52 


7 9 6836 


2 


62 


891827 


1.69 


903008 


4 - 3 1 


094992 


13 


48 


62660 


77934 


796993 
797 i 5o 


2 


62 


891726 


1 -69 


905267 


4-3i 


094733 


12 


I 49 


62683 


77916 


2 


61 


891624 


1 -69 


905526 


4-3i 


094474 


11 


J50 
1 51 


62706 


77897 


797307 


2 


61 


8 9 i523 


i-7° 


905784 


4-3i 


0942 1 6 


10 


62728 


77879 


9-797464 


2 


6 1 


9-891421 


1-70 


9-906043 


4-3i 


10-093937 


9 


! 52 


62751 


77861 


797621 


2 


61 


8gi3i9 


1.70 


906302 


4-3.1 


093698 


8 


| 53 


62774 


77843 


797777 


2 


61 


891217 


1-70 


906560 


4-3i 


093440 


7 


54 


62796 


77824 


797934 


2 


61 


891 11 5 


1-70 


906819 


4-3i 


093 1 81 


6 


55 


62819 


77806 


798091 


2 


61 


891013 


1-70 


907077 


4- 3 1 


092923 


5 


56 


62842 


77788 


798247 


2 


61 


89091 1 


1-70; 


907336 


4-3i 


092664 


4 


57 


62864 


77769 


798403 


2 


60 


890809 


1-70 


907594 


4-3i 


092406 


3 


58 


62887 


7775.1 


798560 


2 


60 


890707 


1-70 


907852 


4-3i 


092148 


2 


59 


62909 


77733 


798716 


2 


60 


890605 


1.70 


9081 1 1 


4-3o 


091889 


1 


1 60 


62932 


77715 


798872 


2- 60 


8go5o3 


1-70 


908369 


4-3o 


09163 1 





1 N. cos. 


N. sine. 


L. cos. 


D.l" 


L. sine. 




L. cot. 


D.l" 


L. tang. 


' 


51° 



SINES AND TANGENTS. 39°. 



57 



Rad. = 1. 




Logarithms.— 


-Radius = 10 10 . 




- 


N. sine. 


N. cos. 


L. sine. 


D.l" 


L. cos. 


D.l" 


L. tang. 


Dl." 


L. cot. 







62932 


77715 


9.798872 


2-60 


9-890503 


1.70 


9-908369 


4-3o 


10-091631 


60 


1 


62955 


77696 


799028 


2- 60 


890400 


I- 71 


908628 


4-3o 


091372 


59 


2 


62977 


77678 


799184 


2- 60 


890298 


I-71 


908886 


4-3o 


091 114 


58 


8 


63ooo 


77660 


799339 


2-59 


890195 


1. 71 


909144 


4-3o 


090856 


57 


4 


63022 


77641 


799495 


2-59 


890093 


I.71 


909402 


4-3o 


090598 


56 


5 


63o45 


77623 


799651 


2-59 


889990 
889888 


1. 71 


909660 


4-3o 


090340 


55 


6 


63o68 


77605 


799806 


2-59 


i- 71 


909918 


4-3o 


090082 
089823 


54 


7 


63090 


77586 


799962 


2-59 


889785 


1. 71 


910177 


4-3o 


53 


8 


63u3 


77568 


800117 


2-59 

2-58 


889682 


1. 71 


910435 


4-3o 


089565 


52 


9 


63i35 


7755o 


800272 


889579 


1. 71 


910693 
910961 


4-3o 


089307 


51 


10 
11 


63 1 58 


77 53i 


800427 


2-58 


889477 


1. 71 


4-3o 


089049 


50 


63 1 80 


775i3 


9-8oo582 


2-58 


9-889374 


I-" 7 2 


9-911209 


4-3o 


10-088791 


49 


12 


632o3 


77494 


800737 


2-58 


889271 


1-72 


91 U67 


4-3o 


088533 


43 


13 


63225 


77476 


800892 


2-58 


889168 


I.72 


911724 


4-3o 


088276 


47 


14 


63248 


77458 


801047 


2-58 


889064 


I.72 


911982 


4-3o 


088018 


46 


15 


63271 


77439 


801201 


2-58 


888961 


1-72 


912240 


4-3o 


087760 


45 


16 


63293 


77421 


8oi356 


2-57 


888858 


1-72 


912498 


4-3o 


087502 


44 


17 


633i6 


77402 


8oi5u 


2-57 


888755 


1-72 


912766 


4-3o 


087244 


43 


18 


63338 


77384 


8oi665 


2-57 


88865i 


1-72 


913014 


4-29 


086986 


42 


19 


63361 


77366 


801819 


2-57 


888548 


1-72 


913271 


4-29 


086729 


41 


20 
21 


63383 
634o6 


77347 


801973 


2-57 


888444 


1.73 


913529 


4-29 


086471 


40 


77329 


9.802128 


207 
2-56 


9-888341 


1.73 


9-913787 


4-29 


10-086213 


39 


22 


63428 


77310 


802282 


888237 


1.73 


914044 


4-29 


o85g56 


38 


23 


6345i 


77292 


802436 


2-56 


888i34 


1.73 


914302 


4-29 


085698 


37 


24 


63473 


77273 


802589 


2-56 


8S8o3o 


1.73 


914560 


4-29 


o8544o 


36 


25 


634q6 


77255 


802743 


2-56 


887926 


1.73 


9U817 


4-29 


o85i83 


35 


26 


635i8 


77236 


802897 


2-56 


887822 


1.73 


915075 


4-29 


084925 


34 


27 


6354o 


77218 


8o3o5o 


2-56 


887718 


1.73 


9i5332 


4-29 


084668 


33 


28 


63563 


77199 
77i8i 


8o32o4 


2-56 


887614 


1.73 


915590 


4-29 


084410 


32 


29 


63585 


8o3357 


2-55 


887510 


1.73 


913847 


4-29 


o84i53 


31 


30 


636o8 


77162 


8o35u 


2-55 


887406 


i-74 


916104 


4-29 


o838 9 6 


30 


31 


6363o 


77*44 


9-8o3664 


2-55 


9-387302 


i-74 


9-916362 


4-29 


io-o83638 


29 


32 


63653 


77125 


8o38i7 


2-55 


887198 


i-74 


916619 


4-29 


oS338i 


28 


33 


63675 


77107 


803970 


2-55 


887093 


i-74 


916877 


4-29 


o83i23 


27 


34 


63698 


77088 


804123 


2-55 


886989 


i-74 


9Hi34 


4-29 


082866 


26 


35 


63720 


77070 


804276 


2-54 


886885 


i-74 


9j73 9 i 


4-29 


082609 


25 


36 


63 7 42 


77o5i 


804428 


2-54 


886780 


i-74 


917648 


4-29 


o82352 


24 


37 


63765 


77033 


8o458i 


2-54 


886676 


1-74 


917903 


4-29 


082095 


23 


38 


j 63787 


770U 


804734 


2-54 


886571 


i-74 


918163 


4-28 


081837 


22 


39 


638io 


76996 


804886 


2-54 


886466 


i-74 


918420 


4-28 


081 58o 


21 


40 


63832 


76977 


8o5o39 


2-04 


886362 


i- 7 5 


918677 


4-28 


o8i323 


20 


41 


63854 


76959 


9-805191 


2-54 


9-886237 


i- 7 5 


1 9-918934 


4-28 


10-081066 


19 


I 42 


63877 


76940 


8o5343 


2-53 


886i52 


i- 7 5 


919191 


4-28 


080809 


18 


43 


63899 


76921 


8o5495 


2-53 


886047 


i- 7 5 


919448 


4-28 


o8o552 


17 


44 


I63922 


76903 
76884 


8o5647 


2-53 


88D942 


i- 7 5 


919705 


4-28 


080295 


16 


45 


163944 


805799 


2-53 


88583 7 


i. 7 5 


919962 


4-28 


o8oo38 


15 


46 


63 9 66 


76866 


805951 


2-53 


885 7 32 


i. 7 5 


920219 


4-28 


079781 


14 


47 


63 9 8 9 


76847 


806 1 o3 


2-53 


885627 


i- 7 5 


920476 


4-28 


079524 


13 


48 


564011 


76828 


806254 


2-53 


885522 


i- 7 5 


920733 


4-28 


079267 


12 


49 


! 64o33 


76810 


806406 


2-52 


885416 


i- 7 5 


920990 


4-28 


079010 


11 


50 
51 


! 64o56 


76791 


8o6557 


2-52 


8853n 


1.76 


921247 
9'92i5o3 


4-28 


078753 


10 


1 64078 


76772 


9-806709 


2-52 


9-885205 


1-76 


4-28 


10-078497 


9 


52 


64100 


76734 


806860 


2-52 


885 1 00 


1.76 


921760 


4-28 


078240 


8 


53 | 64123 


76735 


807011 


2-52 


884994 


1 -76 


922017 


4-28 


077983 


7 


54 


1 64145 


76717 


1 807163 


2-52 


884889 


1-76 


922274 


4-28 


077726 


6 


55 


64167 


76698 


807314 


2-52 


884783 


1.76 
1.76 


922530 


4-28 


077470 


5 


56 


64190 


76679 


807465 


2-5l 


884677 


922787 


4-28 


077213 


4 


57 642 1 2 


76661 


80760 


2-5l 


884572 


1-76 


923044 


4-28 


076956 


3 


58 


64234 ! 76642 


807766 


2-31 


. 884466 


1.76 


9233oo 


4-28 


076700 


2 


59 


64256 


76623 


807917 


2-5l 


88436o 


1.76 


923557 


4-2 7 


076443 


1 


60 


64279 


76604 


808067 


2-5l 


884254 


1.77 


9 238i3 


4-27 


076187 





N. cos 


. N. sine 


.? L. cos. 


D. 1" 


L. sine. 




L. cot. 


D.l" 


L. tang. 


' 


50° 



58 



SINES AND TANGENTS. — 40°. 



Rad.=1. 




LOGARITHMS.- 


-Radius = 10 }O . 




| ' JN.sine 


N. cos 


L. sine. 


D. 1" 


L. COS. 


D.l" 


L. tang. 


D.l" 


L. cot. 







64279 


76604 


9-808067 


2-5l 


9-884254 


'•77 


9-923813 


4-27 


10-076187 


60 


1 


643oi 


76586 


808218 


2-5l 


884148 


1.77 


924070 


4-27 


075930 


59 


2 


64323 


76567 


8o8368 


2-5l 


884042 


'•77 


924327 


4-27 


075673 


58 


8 


64346 


76548 


8o85i 9 


2'5o 


883 9 36 


1-77 


924583 


4-27 


075417 


57 


4 


64368 


7653o 


808669 


2-5o 


88382 9 


1.77 


924840 


4-27 


075160 


56 


5 


64390 


765 1 1 


808819 


2-5o 


883723 


1 -77 


925096 


4-27 


074904 


55 


6 


64412 


76492 


808969 


2-5o 


883617 


'•77 


92535s 


4-27 


074648 


54 


7 


64435 


76473 


8091 19 


2-5o 


8835io 


1-77 


925609 


4-27 


074391 


53 


8 


64457 


76455 


809269 


2-5o 


883404 


1.77 


925865 


4-27 


074i35 


52 


§ 9 


1 64479 


76436 


8094 1 9 


2-49 


883297 


1.78 


926122 


4-27 


073878 


51 


10 !645oi 


76417 


809369 


2-49 


883i 9 i 


1.78 


926378 


4-27 


073622 


50 


11 


64324 


763 9 8 


9-809718 


2-49 


9-883o84 


1.78 


9-926634 


4-27 


I0-073366 


49 


12 


J64546 


7638o 


809868 


2-49 


882977 
882871 


1-78 


926890 


4-27 


073110 


48 


18 


! 64568 


7636i 


810017 


2-49 


..78 


927147 


4-27 


072853 


47 


14 


164590 


76342 


810167 


2-49 


882764 


1.78 


927403 


4-27 


072597 


46 


15 


646 F 2 


7 6323 


8 1 o3 1 6 


2.48 


882657 


1.78 


927659 
927913 


4-27 


072341 


45 


16 


64.635 


76304 


8io465 


2-48 


88255o 


1.78 


4-27 


072085 


44 


17 


6465 7 


76286 


810614 


2.48 


882443 


1-78 


928171 


4-27 


071829 


43 


18 


64679 


76267 


810763 


2-48 


882336 


1.79 


928427 


4-27 


071573 


42 


19 64701 


76248 


8 1 09 ! 2 


2.48 


882229 


1-79 


928683 


4-27 


071317 


41 


1 2 ° 


64723 


76229 


811061 


2.48 


882121 


1.79 


928940 


4-2 7 


071060 


40 


1 ' 21 


64746 


76210 


9-811210 


2-48 


9-882014 


1-79 


9-929196 


4-2 7 


10 070804 


39 


22 |j 64768 76192 


8n358 


2-47 


881907 


1.79 


929452 


4-2 7 


070548 


38 


23 I 64790 


76173 


8n5o7 


2-47 


881799 


1-79 


929708 


4-27 


070292 


37 


1 24 


64812 


76154 


81 1 655 


2-47 


881692 


1-79 


929964 


4-26 


070036 


36 


25 


64834 


76i35 


81 1804 


2-47 


88 1 584 


1-79 


930220 


4-26 


069780 


35 


26 


64856 


76116 


811952 


2-47 


88.477 


1-79 


930475 


4-26 


069525 


34 


3 27 


64878 


76097 


812100 


2-47 


88i36 9 


"79 


930731 


4-26 


069269 


33 


28 


64901 


76078 


812248 


2-47 


88 1 26 1 


i-8o 


930987 


4-26 


069013 


32 


29 


64923 


76059 


812396 


2-46 


881 1 53 


i- 80 


93i243 


4-26 


068757 


31 


30 i 


64945 


76041 


812544 


2-46 


881046 


i- 80 


- 93 1 499 
9-931755 


4-26 


o6H5oi 


30 


M 


64967 


76022 


9-812692 


2-46 


9 -88o 9 38 


"T8o 


~^i6' 


10-068245 


29 


32 64989 


76003 


812840 


2-46 


88o83o 


I- 8o| 


932010 


4-26 


067990 


28 


33 65oti 


73984 


812988 


2-46 


880722 


i- 80 


932266 


4-26 


067734 


27 


34 l]65o33 


7596:") 


8i3i35 


2-46 


88o6i3 


i-8oi 


932522 


4-26 


067478 


26 


35 j 65o55 


7 5 9 46 


8 1 3283. 


2-46 


88o5o5 


i- 80! 


932778 


4-26 


067222 


25 


36 65077 


73927 


8i343o 


2-45 


880397 


i-8o' 


933o33 


4-26 


066967 


24 


37 63100 


75go8 


813578 


2-45 


880289 


1. 81 


933289 


4-26 


0667 1 1 


2:F 


38 65 1 22 


7 588 9 


8i3725 


2-45 


880180 


1 -81! 


933545 


4-26 


o66455 


22 


39 i65i44 


73870 


8 1 38 7 2 


2-45 


880072 


1 -81 j 


9338oo 


4-26 


066200 


21 


40 i; 65!66 


7585 1 
7 583 2~ 


814019 


2-45 


879963 


i-8i 
1. 81! 


934o56 
9-9343 1 1 


4-26 
~4~26 


o65g44 


20 


41 ji65i88 


9-814166 


2-45 


9-879853 


10-065689 


42 652io 


758i3 


8i43i3 


2-45 


879746 


1.8:1 


934567 


4-26 


o65433 


18 


43 65232 


75794 


814460 


2-44 


8 79 63 7 


1,81] 


934823 


4-26 


065177 


17 


44 ! 1652 54 


75775 


814607 


2-44 


879529 


i-8ii 


935078 


4-26 


064922 


16 


45 1 65276 


75756 


8i4753 


2-44 


879420 


i-Sij 


935333 | 4-26 


064667 


15 


46 j 63298 


7 5 7 38 


814900 


2-44 


87931 1 


i-8ij 


935589 I 4-26 


06441 1 


14 


47 65320 


75719 


8i5o46 


2-44 


879202 


1-82' 


935844 4-26 


064 1 56 


13 


48 '65342 


75700 


8i5i 9 3 


2-44 


879093 ' 


1-82; 


936100 | 4-26 


063900 


12 


49 | 653.64 


7568o 


8 1 533 9 


2-44 


878984 


1 -82' 


9.36355 j 4-26 


063645 


11 


50 165386 
fol ||&)4o8 


7566i 
75642 


8 1 5485 


2-43 
2-43 


878875 


1-82, 
1-82 


936610 4-26 
9- 9 36866 4-23 


063390 


10 


9-8r563i 


9-878766 


io-o63i34 


9 


52 j 6543o 


75623 


813778 


2-43 


8 7 8656 


1-82' 


937 1 21 


4-25 


062879 


8 


■v) i; 65432 


756o4 


815924 


2-43 


878347 


I-82 : 


937376 


4-23 


062624 


7 


H li 65474 


75585 


816069 


2-43 


878438 


I -821 


937632 


4-25 


062368 


6 


55 ; 60496 


75566 


816210 


2-43 


878328 


1-82' 


937887 


4-25 


0621 1 3 


5 


06 i 655i8 


75547 


8i636i 


2-43 


878219 


1 -83 


938142 


4-25 


06 1 858 


4 


57 . 6554o 


75528 


816507 


2-42 


878109 


1-83 


938398 4-25 


061602 


3 


58 !j 65562 


735og 


8i6652 


2-42 


877999 j 


i-83 


938653 4-25 


061347 


2 


59 |65584 


75490 


816798 


2-42 


877890 s 


1-83 


938908 ! 4-25 


061092 


1 


60 j656o6 


7 5 47i 


816943 


2-42 


877780 | 


i-83 


939163 4-25 


060837 





i|N. cos. , 


N. sine. 


L. cos. 


D.l"j 


L. sine, j 




L. cot. | I). 1" 


L. tang. 


' 


490 



SINES AND TANGENTS 



59 



Rad.^1. 






LOGARITHMS.- 


—Radius — 10'°. 




' 


N.sine. N. cos. 


L. sine. 


D. 1" 


L. COS. 


D.l" 


| L. tang. 


D.l" 


L. cot. 







60606 J 75471 


9-816943 


2-42 


9.877780 


i-83 


! 9-939163 


4-25 


10-060837 


60 


1 


65628 [75452 


817088 


2-42 


877670 


i-83 


939418 


4-25 


o6o582 


59 


2 


i 6565o | 75433 


817233 


2-42 


877560 


i-83 


939673 


4-25 


060327 


58 


3 


65672 j 75414 


817379 


2-42 


877450 


i-83 


939928 


4-25 


060072 


57 


4 


656o4 1 753 9 5 


817524 


2-41 


877340 


i-83 


94oi83 


4-25 


059817 


56 


5 


65716 i 75375 


817668 


2-41 


877230 


1.84 


940438 


4-25 


059562 


55 


6 


!65 7 38 7 5356 


817813 


2-41 


877120 


1.84 


940694 


4-25 


059306 


54 


7 


; 65759 


75337 


817958 


2-41 


877OIO 


1-84 


940949 


4-25 


05905 1 


53 


8 


|65 7 8i 


7 53i8 


8i8io3 


2-41 


876899 


1.84 


941204 


4-25 


058796 


52 


9 


658o3 


75299 


818247 


2-41 


876789 


1.84 


94U58 


4-25 


o58542 


51 


10 

ir 


! 65825 75280 


818392 


2-41 


876678 


1-84 


941714 
9-941968 


4-25 


o58286 


50 


65847 7526i 


9 -8i8536 


2-40 


9-876568 


1.84 


4-25 


io-o58o32 


49 


12 


65869 75241 


818681 


2-40 


876457 


1-84 


942223 


4-25 


o57777 


48 


13 


' 65891 1 75222 


8i8825 


2-40 


876347 


1-84 


942478 


4-25 


057522 


47 


14 ] 659i3 ! 7D2o3 


818969 


2-40 


876236 


1-85 


942733 


4-25 


057267 


46 


15 


i 60935 


75i84 


819113 


2-40 


876125 


i-85 


942988 


4-25 


057012 


45 


16 


60956 


75i65 


819257 


2-40 


876014 


i-85 


943243 


4-25 


056757 


44 


17 


65 97 8 


75146 


819401 


2-40 


875904 


i-85 


943498 


4-25 


o565o2 


43 


18 


! 66000 


75126 


819045 


2-39 


875793 


i-85 


943752 


4-25 


056248 


42 


19 


66022 75107 


819689 


2-39 


8 7 5682 


i-85 


944007 


4-25 


055993 


41 


20 


1 66044 75o88 


819832 


2-39 


875571 


i-85 


944262 


4-25 


o55738 


40 


21 


i 66066 


75069 


9-819976 


2-39 


9-875459 
875348 


i-85 


9-9445i7 


4-25 


io-o55483 


39 


22 


66088 


75o5o 


820120 


2-39 


i-85 


944771 


4-24 


055229 


38 


23 


i 66109 


75o3o 


820263 


2-39 


875237 


i-85 


945026 


4-24 


054974 


37 


24 


66 i 3i 


75oi 1 


820406 


2-39 


875l26 


i-86 


94528i 


4-24 


054719 


36 


25 !|66i53 


74992 


82o55o 


2-38 


875014 


i-86 


945535 


4-24 


054460 


35 


26 


66175 


74973 


820693 


2-38 


874903 


1.86 


945790 


4-24 


054210 


34 


27 


66197 


74953 


820836 


2-38 


87479 1 


i-86 


946045 


4-24 


o53955 


33 


28 


j 66218 


74934 


820979 


2-38 


874680 


i-86 


946299 


4-24 


053701 


32 


29 


66240 


749i5 


821 122 


2-38 


874568 


i-86 


946554 


4-24 


o53446 


31 


30 


66262 


74896 


821265 


2-38 


8 7 4456 


i-86 


946808 


4-24 


053192 


30 


31 


166284 74876 


9-821407 


2-38 


9-874344 


i-86 


9-947063 


4-24 


10-052937 


29 


32 


! 663o6 j 74857 


82i5oo 


2-38 


874232 


1-87 


9473i8 


4-24 


052682 


28 


33 


66327 74838 


821693 


2-3 7 


874I2I 


1-87 


947572 


4-24 


052428 


27 


84 


1 66349 748i8 


82i835 


2-3 7 


874009 


1-87 


947826 


4-24 


052174 


26 


85 


66371 (747Q9 


821977 


2-3 7 


873896 


1.87 


948081 


4-24 


051919 


25 


36 


663g3 74780 


822120 


2-3 7 


873784 


1-87 


948336 


4-24 


o5i664 


24 


37 


66414 74760 


822262 


2-3 7 


873672 


1-87 


948590 


4-24 


o5i4io 


23 


38 


1 66436 74741 


822404 


2-3 7 


873560 


1-87 


948844 


4-24 


00 1 1 56 


22 


89 


! 66458 74722 


822546 


2.37 


873448 


1-87 


949099 


4-24 


050901 


21 


40 


: 66480 


747o3 
74683 


822688 
9 -822830" 


2-36 


873335 


1.87 
1.87 


949353 


4-24 


o5o647 


20 


41 


665oi 


2-36 


9'873223 


9-949607 


4-24 


io-o5o393 


19 


42 


66523 


74664 


822972 


2-36 


8731 10 


i-88 


949862 


4-24 


o5oi38 


18 


43 


! 66545 


74644 


823ri4 


2-36 


872998 
872885 


i-88 


950116 


4-24 


049884 


17 


44 


66566 74625 


823255 


2-36 


i-88 


950370 


4-24 


049630 


IS | 


45 ! 66588 i 74606 


823397 


2-36 


872772 


i-88! 


95o625 


4-24 


049375 


15 | 


4« i 66610 74586! 


823539 


2-36 


872609 


i-88| 


9O0879 


4-24 


0491 21 


14 j 


1-7 


: 66632 ! 74067 ! 


82368o 


2-35 


872547 


i-88| 


9 5ii33 


4-24 


048867 


13 I 


4S 


'66653! 74548 i 


823821 


2-35 


872434 


1 -88; 


95i388 


4-24 


048612 


12 j 


49 


66675 74528 j 


823 9 63 


2-35 


872321 


1 -881 


951642 


4-24 


048358 


11 | 


50 


: 66697 


745o 9 


824104 


2-35 


872208 

9-872095 


i-88! 


951896 


4-24 


048104 


10 [ 


51 |j 66718 


74489 \ 


9-824245 


2-35 


9-952i5o 


4-24 


io-o4785o 


9 f 


52 166740174470 


824386 


2-35 


871981 


i-8 9 


9524o5 


4-24 


047595 


8 j 


53 66762 


7445i 


824527 


2.35 


871868 


..89 


952659 


4-24 


047341 


7 


54 166783 


7443 1 


824668 


2-34 


871755 


1.89 


952913 


4-24 


047087 


6 


55 ! 668o5 


74412 


824808 


2-34 


871641 


1.89 


953167 


4-23 


o46833 


5 


56 | 66S27 


74392 


824949 


2-34 


871528 


i-8 9 


953421 


4-23 


046579 


4 


57 | 66848 


743 7 3 


825090 


2-34 


871414 


1-89 


953675 


4-23 


046325 


3 


58 i: 66870 


74353 


82523o 


2-34 


871301 


1.89 


953929 


4-23 


046071 


2 


59 1(66891 | 74334 


825371 


2-34 


871187 


,-8 9 


954i83 


4-23 


045817 


1 


60 


j 66913 743i4 


8255u 


2-34 


871073 


1-90 


954437 4-23 


045563 







N. cos. N. sine.] 


L- cos. ! 


D. 1" 


L. sine. 




L. cot. J D. 1" 


L. tang. 


' 


4§ c 



60 



SINES AND TANGENTS* 42°. 



Rad. = 1. 


1 




Logarithms.— 


-Ramus = 10 10 . 




' 


N. sine. 


N. cos. 


L. sine. 


D.l" 


L. cos. 


D.l" 


L. tang. 


D 


.'■¥'. 


L. cot. 







66913 


743i4 


9-8255u 


2 


34 


9-871073 


1-90 


9-954437 


4 


23 


io-o45563 


60 


1 


66 9 35 


74295 


82565i 


2 


33 


870960 


1 -90 


904691 


4 


23 


045309 


59 


2 


66 9 56 


74276 


825791 


2 


33 


870846 


1-90 


954945 


4 


23 


o45o55 


58 


3 


66978 


74256 


825931 


2 


33 


870732 


1-90 


955200 


4 


23 


044800 


57 


4 


66999 


7423 7 


826071 


2 


33 


870618 


1 -90 


955454 


4 


23 


044546 


56 


5 


67021 


74217 
74198 


826211 


2 


33 


870504 


1-90 


955707 


4 


23 


044293 


55 


6 


67043 


82635i 


2 


33 


870390 


1-90 


955961 


4 


23 


044039 


54 


7 


67064 


74178 


826491 


2 


33 


870276 


1-90 


9562i5 


4 


23 


043785 


53 


8 


67086 


74i59 


82663i 


2 


33 


870161 


1-90 


956469 


4 


23 


o4353i 


52 


9 


67107 


74i39 


826770 


2 


32 


870047 


1-91 


956723 


4 


23 


043277 


51 


10 


67129 


74120 


826910 


2 


32 


869933 


1-91 


956977 


4 


23 


o43o23 


50 


11 


67151 


74100 


9-827049 


2 


32 


9-869818 


1-91 


9-957231 


4 


23 


10-042769 


49 


12 


67172 


74080 


827189 


2 


32 


869704 


1-91 


957485 


4 


23 


0425i5 


48 


IS 


67194 


74061 


827328 


2 


32 


869589 


1. 91 


957739 


4 


23 


042261 


47 


14 


67215 


74041 


827467 


2 


32 


86 Q 474 


1-91 


957993 


4 


23 


042007 


46 


15 


67237 


74022 


827606 


2 


32 


86 9 36o 


1-91 


958246 


4 


23 


041754 


45 


16 


67258 


74002 


827745 


2 


32 


869245 


1-91 


9585oo 


4 


23 


041 5oo 


44 


17 


67280 


73983 


827884 


2 


3i 


869130 


1-91 


958754 


4 


23 


041246 


43 


18 


67301 


73963 


828023 


2 


3i 


86 9 oi5 


1-92 


959008 


4 


23 


040992 


42 


19 


6 7 323 


73944 


828162 


2 


3i 


868900 


1-92 


959262 


4 


23 


040738 


41 


20 


67344 


73924 


8283oi 


2 


3i 


868 7 85 


1-92 


959516 


4 


23 


040484 


40 
39 


21 


67366 


73904 


9-828439 


2 


3i 


9-868670 


1-92 


9-959769 


4 


23 


IO- 04023 1 


22 


6 7 38 7 


73885 


828578 


2 


3i 


868555 


1-92 


960023 


4 


23 


039977 


83 


23 


67409 


73865 


828716 


2 


3i 


868440 


1-92 


960277 


4 


23 


039723 


37 


24 


67430 


73846 


828855 


2 


3o 


868324 


1-92 


96o53i 


4 


23 


039469 


36 


25 


67452 


73826 


828993 


2 


3o 


868209 


1.92 


960784 


4 


23 


039216 


35 


26 


67473 


73806 


829131 


2 


3o 


868o 9 3 


1-92 


961038 


4 


23 


038962 


34 


27 


67495 


73787 


829269 


2 


3o 


867978 
867862 


i- 9 3 


961 291 


4 


23 


038709 
o3&455 


33 


28 


6 7 5i6 


73767 


829407 


2 


3o 


i- 9 3 


961545 


4 


23 


32 


29 


6 7 538 


73747 


829545 


2 


3o 


867747 


i- 9 3 


961799 
962052 


4 


23 


o382oi 


31 


30 


67559 


73728 


829683 


2 


3o 


867631 


i- 9 3 


4 


23 


037948 


30 


31 


67580 


73708 


9-829821 


2 


29 


9-86 7 5i5 


i- 9 3 


9-962306 


4 


23 


10-037694 


29 


32 


67602 


7 3688 


829909 


2 


29 


86 7 3 99 
867283 


i- 9 3 


962560 


4 


23 


037440 


28 


33 


67623 


73669 


830097 


2 


29 


i. 9 3 


962813 


4 


23 


037187 


27 


34 


67645 


73649 


830234 


2 


29 


867167 


T.. 9 3 


963067 


4 


23 


036933 


26 


35 


67666 


73629 


83o372 


2 


29 


867051 


i- 9 3 


963320 


4 


23 


o3668o 


25 


| 36 


67688 


736io 


83o5o9 


2 


29 


866 9 35 
866819 


i-94 


963574 


4 


23 


036426 


24 


37 


67709 


73590 


83o646 


2 


29 


i-94 


963827 


4 


23 


036173 


28 


38 


67730 


73570 


830784 


2 


29 


866703 


1-94 


964081 


4 


23 


03D919 
035665 


22 


39 


67752 


7355i 


830921 


2 


28 


866586 


1-94 


964335 


4 


23 


21 


40 


67773 


7353i 


83io58 


2 


28 


866470 


i- 9 4 


964588 


4 


22 


o354i2 


20 


41 


67795 


735n 


9-83ii95 


2 


28 


9-866353 


1-94 


9-964842 


4 


22 


io-o35i58 


19 


42 


67816 


73491 


83i332 


2 


28 


866237 


i- 9 4 


965095 


4 


22 


o34go5 


18 


43 


6 7 83 7 


73472 


831469 


2 


28 


866120 


1-94 


965349 


4 


22 


o3465i 


17 


44 


67859 


73452 


83 1 606 


2 


28 


866004 


i- 9 5 


965602 


4 


22 


034398 


16 


45 


67880 


73432 


831742 


2 


28 


865887 


i- 9 5 


9 65855 


4 


22 


o34U5 


15 


46 


67901 


734i3 


83 J 879 
832oi5 


2 


28 


865770 
865653 


i- 9 5 


966109 


4 


22 


033891 


14 


47 


67923 


733o3 


2 


27 


i- 9 5 


966362 


4 


22 


033638 


13 


48 


67944 


7 33 7 3 


832i52 


2 


27 


865536 


1.96 


966616 


4 


22 


o33384 


12 


49 


67965 


73353 


832288 


2 


27 


865419 


i- 9 5 


966869 


4 


22 


o33i3i 


11 


50 


67987 


73333 


832425 


2 


27 


8653o2 


i- 9 5 


967123 


4 


22 


032877 


10 


51 


68008 


733U 


9 -83256i 


2 


27 


9-865i85 


i- 9 5 


9-967376 


4 


22 


io-o32624 


9 


52 


1 68029 


73294 


832697 


2 


27 


865o68 


i- 9 5 


967629 
967883 


4 


22 


032371 


8 


53 


68o5i 


73274 


832833 


2 


27 


864950 


i- 9 5 


4 


22 


032117 


7 


54 


68072 


7 3254 


832969 


2 


26 


864833 


1-96 


968i36 


4 


22 


o3i864 


6 


J 55 


68093 


73234 


833 1 o5 


2 


26 


864716 


1-96 


968389 


4 


22 


o3i6u 


5 


I 56 


68u5 


732i5 


833241 


2 


26 


864598 
864481 


1.96 


968643 


4 


22 


o3i357 


4 


57 


68i36 


73195 


8333 77 


2 


26 


I.96 


968896 


4 


22 


o3no4 


3 


58 


68i5 7 


73175 


8335i2 


2 


26 


864363 


1.96 


969149 


4 


22 


o3o85i 


2 


59 


68179 


73 1 55 


833648 


2 


26 


864245 


1.96 


969403 


4 


22 


o3o597 


1 


60 


68200 


73i35 


833 7 83 


2-26 


864127 


1.96 


969656 


4-22 


o3o344 







|N. cos.jN.6ine. 


L. cos. 


D.l" 


L. sine. 




L. cot. 


D.l" 


L. tang. 


' 


4?° 



SINES AND TANGENTS. — 43 c 



61 



Rad. = 1. 






Logarithms.— 


■Radius = 10 10 . 




' 


N.sine. 


N. cos. 


L. sine. 


D. 1" 


L. cos. 1 


3.1" 


L. tang. 


D. 1" 


L. cot. 







68200 


7 3i35 


9-833783 


2 


26 


9-864127 i 


.96 


9-969656 


4-22 


io-o3o344 


60 


1 


68221 


73n6 


833919 





25 


864010 i 


.96 


969909 


4-22 


030091 
029838 


59 


2 


68242 


73096 


834o54 


2 


25 


8638 9 2 i 


•97 


970162 


4> 22 


5S 


8 


68264 


73076 


834189 
834325 


2 


25 


863774 


•97 


970416 


4-22 


029584 


57 


4 


68280 


73o56 


2 


25 


863656 


•97 


970669 


4-22 


029331 


56 


5 


683o6 


73o36 


83446o 


2 


2 5 


863538 


•97 


970922 


4-22 


029078 


55 


6 


6832 7 


73oi6 


834595 

83473o 


2 


25 


863419 


•97 


971175 


4-22 


028825 


54 


7 


68349 


72996 


2 


25 


8633oi 


•97 


971429 


4-22 


028571 


53 


8 


683 7 o 


72976 


834865 


2 


25 


863 1 83 


•97 


971682 


4-22 


0283i8 


52 


9 


683qi 


72957 


834999 


2 


24 


863o64 


•97 


971935 


4-22 


028065 


51 


10 
11 


68412 


72937 


835i34 


2 


24 


862946 


.98 


972188 


4-22 


027812 


50 
49 


68434 


72917 


9-835269 


2 


24 


9.862827 


.98 


9-97244I 


4-22 


10-027559 


12 


68455 


72897 


8354o3 


2 


24 


862709 


.98 


972694 


4-22 


027306 


48 


13 


68476 


72877 


835538 


2 


24 


862590 


.98 


972948 


4-22 


027052 


47 


14 


68497 


72857 


835672 


2 


24 


862471 


.98 


973201 


4-22 


026799 


46 


15 


685i8 


72837 


835807 


2 


24 


862353 


• 9 8 


973454 


4-22 


026546 


45 


16 


68539 


72817 


835941 


2 


24 


862234 


.98 


973707 


4-22 


026293 


44 


17 


6856i 


72797 


836075 


2 


23 


862ii5 


.98 


973960 


4-22 


026040 


43 


18 


68582 


72777 


836209 
83634^ 


2 


23 


861996 
861877 


.98 


974213 


4-22 


025787 


42 


19 


686o3 


72757 


2 


23 


.98 


974466 


4-22 


025534 


41 


20 


68624 


72737 


836477 


2 


23 


86i 7 58 


^99 


974719 


4-22 


025281 


40 


21 


68645 


72717 


9-8366i 1 


2 


23 


9- 861 638 


•99 


9-974973 


4'22 


I0-025027 


39 


22 


68666 


72697 


836745 


2 


23 


861519 


.99 


975226 


4-22 


024774 


38 


23 


68688 


72677 


8368 7 8 


2 


23 


861400 


■99 


975479 


4-22 


024521 


37 


24 


68709 


72657 


837012 


2 


22 


861280 


•99 


975732 


4-22 


024268 


36 


25 


68730 


72637 


837146 


2 


22 


861161 


.99 


975985 


4-22 


024015 


35 


26 


68751 


72617 


837279 


2 


22 


861041 


•99 


976238 


4-22 


023762 


34 


27 


68772 


72597 


837412 


2 


22 


860922 
860802 


•99 


97 6 49i 


4-22 


023509 


33 


28 


68793 


72577 


837546 


2 


22 


•99 


976744 


4-22 


023256 


32 


29 


68814 


72557 


837679 


2 


22 


860682 : 


•00 


976997 


4-22 


o23oo3 


31 


30 


68835 


72537 


83 7 8i2 


2 


22 


86o562 : 


•00 


977250 


4-22 


022750 


30 


31 


688D7 


72517 


9-837945 


2 


22 


9 • 860442 : 


•00 


9-9775o3 


4-22 


10-022497 


29 


32 


68878 


72497 


838078 


2 


21 


86o322 5 


•00 


977756 


4-22 


022244 


28 


33 


68899 


72477 


8382 1 1 


2 


21 


860202 : 


•00 


978009 


4-22 


021991 
021738 


27 


34 


68920 


72457 


838344 


2 


21 


860082 : 


•00 


978262 


4-22 


26 


35 


68941 


72437 


838477 


2 


21 


859962 : 


•00 


9785i5 


4-22 


021485 


25 


36 


68962 


72417 


8386io 


2 


21 


859842 2 


•00 


978768 


4-22 


021232 


24 


37 


68 9 83 


72397 


838742 


2 


21 


859721 1 


•01 


979021 


4'22 


020979 


23 


38 


69004 


72377 


838875 


2 


21 


859601 2 


•01 


979274 


4-22 


020726 


22 


39 


69025 


72357 


839007 


2 


21 


859480 : 


•01 


979527 


4'22 


020473 


21 


40 


69046 


72337 


839140 


2 


20 


859360 2 


•01 


979780 


4-22 


020220 


20 


41 


69067 


723i7 


9-839272 


2 


20 


9-859239 ; 


•01 


9-980033 


4-22 


10-019967 


19 


42 


69088 


72297 


839404 


2 


20 


8591 19 5 
858998 2 
858877 2 


•01 


980286 


4-22 


0I97I4 


18 


43 


69109 


72277 
72257 


83 9 536 


2 


20 


•01 


98o538 


4'22 


OI9462 


17 


44 


69130 


83 9 668 


2 


20 


•01 


980791 


4-21 


OI9209 


16 


45 


6gi5i 


72236 


839800 


2 


20 


858756 5 


•02 


981044 


4-21 


oi8 9 56 


15 


46 


69172 


72216 


839932 


2 


20 


858635 5 


•02 


981297 


4-21 


018703 


14 


47 


69193 


72196 


840064 


2 


J 9 


8585i4 2 


•02 


98i55o 


4-21 


oi845o 


13 


48 


69214 


72176 


840196 


2 


19 


8583 9 3 2 


•02 


981803 


4-21 


018197 


12 


49 


69235 


72i56 


840328 


2 


*8 


858272 2 
858i5i 2 


•02 


982056 


4-21 


017944 


11 


50 


69256 


72i36 


840459 


2 


19 


• 02 


982309 


4-21 


017691 


10 


51 


69277 


72116 


9-840591 


2 


19 


9- 858029 2 
857908 2 


•02 


9-982562 


4-21 


10-017438 


9 


52 


69298 


72095 


840722 


2 


19 


•02 


982814 


4-21 


017186 


8 


53 


69319 


72075 


84o854 


2 


19 


857786 2 


•02 


983067 


4-21 


016933 


7 


54 


69340 


72o55 


840985 


2 


;i 


857665 2 


•o3 


983320 


4-21 


016680 


6 


55 


69361 


72o35 


841116 


2 


857543 2 


• o3 


983573 


4-21 


016427 


5 


56 


6 9 382 


72015 


841247 


2 


18 


857422 2 


• o3 


983826 


4-21 


016174 


4 


57 


69403 


71995 


841378 


2 


18 


857300 2 


• o3 


984079 


4-21 


015921 


3 


58 


69424 


71974 


841509 


2 


18 


857178 2 


•o3 


984331 


4-21 


015669 


2 


59 


69445 


71954 


841640 


2 


18 


857056 2 


-o3 


984584 


4-21 


oi54i6 


1 


60 


69466 
N. cos. 


71934 


841 77 1 


2 


18 


856 9 34 2 


•o3 


984837 


4-21 


oi5i63 





N. sine. 


L. cos. 


D 


V 


L. sine. 




L. cot. 


D. 1" 


L. tang. 


' 


46° 



62 



SINES AND TANGENTS. 



Rad. = 1. 


Logarithms. — Radius = IO 10 . 




1 

2 

S 

4 

5 

6 

7 

8 

9 

10 

11 

12 

13 

14 

15 

16 

17 

18 

19 

20 


N. sine. 


N. cos. 


L. sine. 


D 


1" 


L. cos. 


D 


1" 


L. tang. 


Dl." 


L. cot. 




69466 
69487 
69608 
69529 
69549 
69570 
69591 
69612 
6 9 633 
69654 
69675 


7i9 3 4 
71014 
71894 
71873 
7i853 
7i833 
71813 
71792 
71772 
71752 
71732 


9-841771 
841902 
842033 
842163 
842294 
842424 
842555 
842685 
8428i5 
842946 
843076 


2 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 


18 
18 
18 
17 
17 
17 
17 
17 
17 

17 


9-856 9 34 
8568i2 
856690 
856568 
856446 
856323 
856201 
856078 
855 9 56 
855833 
8557H 


2 
2 
2 
2 
2 
2 
2 
2 
2 
2 
2 


o3 
o3 
04 
04 
04 
04 
04 
04 
04 
04 
o5 


9-984837 
986090 
985343 
985596 
985848 
986101 
986354 
986607 
' 986860 
9871 1 2 
987365 


4 
4 
4 
4 
4 
4 
4 
4 
4 
4 
4 


21 
21 
21 
21 
21 
21 
21 
21 
21 
21 
21 


io-oi5i63 
014910 
014657 
014404 
oi4i52 
013899 
01 3646 
oi3393 
oi3i4o 
012888 
012635 


60 
59 
58 
57 
56 
55 
54 
53 
52 
51 
50 


69696 
69717 
69737 
69758 
69779 
69800 
69821 
69842 
69862 
6 9 883 


71711 
71691 
71671 
7i65o 
7i63o 
71610 
71590 
71569 
7i549 
71529 


9-843206 
843336 
843466 
8435 9 5 
843725 
843855 
843984 
8441 U 
844243 
844372 


2 
2 
2 
2 
2 
2 
2 
2 
2 
2 


16 
-l6 
16 
16 
16 
16 
16 

i5 
i5 
i5 


9-855588 
855465 
855342 
855219 
855o 9 6 
854073 
85485o 
854727 
8546o3 
85448o 


2 
2 
2 
2 
2 
2 
2 
2 
2 
2 


o5 

o5 
o5 
o5 
o5 
o5 
o5 
06 
06 
06 


9-987618 
98787I 
988123 
988376 
988629 
988882 
989134 
989387 
989640 
989893 


4 
4 
4 
4 
4 
4 
4 
4 
4 
4 


21 
21 
21 
21 
21 
21 
21 
21 
21 
21 


10-012382 

012129 
01 1877 
011624 
011371 
011118 
010866 
oio6i3 
oio36o 
010107 


49 
48 
47 
46 
45 
44 
43 
42 
41 
40 


21 
22 
23 
24 
25 
26 
27 
28 
29 
30 
31 
32 
33 
34 
35 
36 
37 
38 
39 
40 

41 

42 
43 
44 

45 
46 
47 
48 
49 
50 


69904 
69926 
69946 
69966 
69987 
70008 
70029 
70049 
70070 
70091 


7i5o8 
71488 
71468 
71447 
71427 
71407 
7i386 
7 1 366 
7i345 
7i325 


9- 844502 
844631 
844760 
844889 
845oi8 
845i47 
845276- 
8454o5 
845533 
845662 


2 
2 
2 
2 
2 
2 
2 
2 
2 
2 


i5 
i5 
i5 
i5 
i5 
i5 
14 
14 
14 
14 


9-854356 
854233 
854109 
853 9 86 
853862 
853738 
853614 
853490 
853366 
853242 


2 
2 
2 
2 
2 
2 
2 
2 
2 
2 


06 
06 
06 
06 
06 
06 
07 
07 

07 
07 


9-990I45 
990398 
99065 I 
990903 

991 1 56 
991409 
991662 
991914 
992167 
992420 


4 

4 
4 
4 
4 
4 
4 
4 
4 
4 


21 
21 
21 
21 
21 
21 
21 
21 
21 
21 


10-009855 
009602 
009349 
009097 
008844 
008591 
oo8338 
008086 
007833 
007580 


39 
38 
37 
36 
35 
34 
33 
32 
31 
30 

29 
28 
27 
26 
25 
24 
23 
22 
21 
20 


70112 
70132 
70i53 
70174 
70195 
702i5 
70236 
70237 
70277 
70298 


7i3o5 
71284 
71264 
71243 
71223 
71203 
71182 
71162 
71141 
71121 


9-845790 
845919 
846047 
846175 
8463o4 
846432 
84656o 
846688 
846816 
846944 


2 
2 
2 
2 
2 
2 
2 
2 
2 
2 


14 
U 
14 
14 
14 
i3 
i3 
i3 
i3 
i3 


9-853n8 
852994 
852869 
852745 
852620 
852496 
852371 
852247 

852122 

851997 


2 
2 
2 
2 
2 
2 
2 
2 
2 
2 


07 
07 
07 
07 
07 
08 
08 
08 
08 
08 


9-992672 
992925 
993178 
993430 
993683 
993936 
994189 
994441 
994694 
994947 


4 
4 
4 
4 
4 
4 
4 
4 
4 
4 


21 
21 
21 
21 
21 
21 
21 
21 
21 
21 


10-007328 
007075 
006822 
006570 
006317 
006064 
oo58ii 
oo5559 
oo53o6 
oo5o53 


70319 
70339 
7o36o 
70381 
70401 
70422 
70443 
70463 
70484 
7o5o5 


71 100 
71080 
71009 
71039 
71019 
70998 
70978 
70907 
70937 
70916 


9-847071 
847199 
847327 
847454 
847582 

847709 
847836 
847964 
848091 
848218 


2 
2 
2 
2 
2 
2 
2 

2 
2 


i3 
i3 
i3 
12 
12 
12 
12 
12 
12 
12 


9-851872 
851747 
85i622 
85 1 497 
85i372 
85i246 
85ii2i 
850996 
850870 
850745 


2 
2 
2 
2 
2 
2 
2 
2 
2 
2 


08 
08 
08 
09 
09 
09 
09 
09 
09 
09 


9-995199 
995452 
995705 

99 5 9 5 7 
996210 
996463 
996715 
996968 
997221 
997473 


4 
4 
4 
4 
4 
4 
4 
4 
4 
4 


21 
21 
21 
21 
21 
21 
21 
2 I 
21 
21 


10-004801 
004548 
004295 
004043 
003790 
oo3537 
oo3285 
oo3o32 
002779 
002527 


19 
18 
17 
16 
15 
14 
13 
12 
11 
10 


51 
52 
53 
54 
55 
56 
57 
58 
59 
60 


70523 

70046 
70567 
7 o58 7 
70608 
70628 
70649 
70670 
70690 
70711 


70896 
70875 
70855 
70834 
70813 
70793 
70772 
70752 
70731 
707 1 1 


9-848345 
848472 
848599 
848726 
848852 
848979 
849106 
849232 
849309 
849486 


2 
2 
2 
2 
2 
2 
2 
2 
2 
2 


12 
11 
11 
11 
11 
11 
11 
11 
11 
11 


9-850619 
85o493 
85o368 
850242 
85ou6 
849990 
849864 
849738 
84961 1 
849485 


2 
2 
2 
2 
2 
2 
2 
2 
2 
2 


09 
10 
10 
10 
10 
10 
10 
10 
10 

,0 


9.997726 

997979 
998231 
998484 
998737 
998989 
999242 
999495 

999747 
1 • 000000 


4 
4 
4 
4 
4 
4 
4 
4 
4. 
4 


21 
21 
21 
21 
21 
21 
21 
21 
21 
21 


10-002274 
002021 
001769 
ooi5io 
001263 

OOIOII 

000758 

ooo5o5 

ooo253 

1 • 000000 


9 
8 
7 
6 
5 
4 
3 
2 
1 



N. cos. 


N. sine. 


L. cos. 


D. 1" 


L. sine. 




L. cot. 


D.l" 


L. tang. 


' 


45° 



TABLE III 



CONTAINING 



LATITUDES AND DEPARTURES 



DEGREES AND FOURTHS, 



AND DISTANCES FROM 1 TO 10. 



64: 



LATITUDES AND DEPARTURES. 





Dist. 1. 


Dist. 2. 


Dist. 3. 


Dist. 4. 


Dist. 5. 




Bear. 












Bear. 

89% 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


oH 


1 • 0000 


• 0044 


2 • 0000 


0-0087 


3-oooo 


o-oi3i 


4-oooo 


0-0175 


5-oooo 


0-02 l8 


oH 


0000 


0087 


1-9999 


0175 


2-9999 


0262 


3-9998 


0349 


4-9998 


0436 


8q^ 


OK 


0-9999 


oi3i 


9998 


0262 


9997 


0393 


9997 


o524 


9996 


o654 


893^ 


1° 


9998 


0175 


99Q7 


o349 


9995 


o524 


9994 


0698 


9992 
9988 


0873 


89° 


iK 


9998 


02I& 


999 5 


0436 


9993 


06 54 


9990 


0873 


1091 


88K 


i% 


9997 


0262 


999 3 


o524 


9990 


0785 


9986 


1047 


99 83 


1 309 


88>£ 


>K 


9995 


o3o5 


9991 


061 1 


9986 


0916 


9981 


1222 


9977 


1627 


88* 


2° 


-9994 


0349 


9988 


0698 


9982 


1047 


9976 


1396 


9970 


H45 


88° 


2% 


9992 


0393 


99 85 


0785 


9977 


1178 


9969 


1570 


9961 


1963 


87K 


2X 


9990 
0-9988 


0436 


9981 


0872 


9971 


1 309 


9962 


1745 


99 52 


2181 


87* 


2% 


0-0480 


1-9977 


0-0960 


2-9965 


0-1439 


3-9954 


0-1919 


4-9942 


0-2399 


87* 


3° 


9986 


0D23 


9973 


1047 


99 5 9 


1570 


9945 


2093 


99 3j 


2617 


87° 


3K 


9984 


o56 7 


9968 


1 1 34 


99 52 


1701 


9936 


2268 


9920 


2835 


865/ 


m 


9981 


06 ro 


9963 


1221 


9944 


i83i 


9925 


2442 


9907 


3o5o 


86* 


3K 


9979 


o654 


99 5 7 


j3o8 


9936 


1962 


9914 


2616 


9 8 9 3 


3270 


86* 


4° 


9976 


0698 


9 9 5i 


1395 


9927 


2093 


9903 


2790 


9878 


3488 


86° 


AK 


99-3 


0741 


9945 


1482 


9918 


2223 


9890 


2964 


9 863 


3 7 o5 


85 % 


4% 


9969 


0785 


9938 


1569 


9908 


2354 


9877 


3i38 


9846 


3923 


85^ 


4K 


9966 


0828 


99 3 ' 


1 656 


9897 


2484 


9863 


33i2 


9828 


4i4o 


85* 


5° 


9962 


0872 


9924 


1743 


9886 


26i5 


9848 


3486 


9810 


4358 


85° 


5K 


0-9958 


0-0915 


1-9916 


o- i83o 


2-9874 


0-2745 


3- 9 832 


o-366o 


4-9790 


0-4570 


84K 


5% 


9954 


0958 


9908 


1917 


9862 


2875 


9816 


3834 


9770 


479 2 


84* 


5K 


9960 


1002 


9899 


2004 


9849 


3 006 


9799 


4008 


9748 


5009 


84'^ 


6° 


9945 


1045 


9890 


2091 


9 836 


3i36 


9781 


4181 


9726 


5226 


840 


bK 


9941 


1089 


9881 


2177 


9822 


3266 


9762 


4355 


9703 


5443 


83K 


b% 


9936 


Il32 


9871 


2264 


9807 


33 9 6 


9743 


4528 


9679 
9 653 


566o 


83* 


bK 


9 9 3 1 


1 1 75 


9861 


235i 


9792 


3526 


9723 


4701 


58 77 


83* 


7° 


992D 


1219 


985i 


243 7 


9776 


3656 


9702 


48 7 5 


9627 


6093 


830 


7^ 


9920 


1262 


9840 


2524 


9760 


3786 


9680 


5o48 


9600 


63io 


82K 


7X 


9914 


i3o5 


9829 


261 1 


9743 


3916 


96D8 


5221 


9 5 7 2 


6526 


82* 


IK 


0-9909 
9903 


0-1349 


1-9817 


0-2697 


2.9726 


0.4046 


3.9635 


0-5394 


4.9543 


0-6743 


81H 


8^ 


i3g2 


9 8o5 


2783 


9708 


4n5 


9611 


556 7 


95i3 


6959 


82° 


SM 


9897 


1435 


9793 


2870 


9690 


43o5 


9 586 


5740 


9483 


7175 


81K 


8% 


9890 
9884 


1478 


978o 


2956 


9670 


4434 


956i 


5912 


9 45i 


7 3 9 o 


81* 


8K 


l52I 


9767 


3o42 


9 65 j 


4564 


9534 


6o85 


9418 


7606 


81K 


9 3 


9877 


1 564 


9754 


3129 


9631 


4693 


95o8 


6257 


9384 


7822 


81° 


gK 


9870 


1607 


9740 


32i5 


9610 


4822 


9480 


643o 


935o 


8037 


80K 


9^ 


9863 


i65o 


9726 


33oi 


9589 


495i 


945 1 


6602 


93i4 


8252 


8o# 


9K 


98D6 


1693 


9711 


338 7 


9567 


5o8o 


9422 


6774 


9278 


8467 


80^ 


10° 


9848 


1736 


9696 


3473 


9544 


5209 


9 3 9 2 


6946 


9240 


8682 


80° 


10* 


0-9840 


0-1779 


1-9681 


0-3559 


2-9521 


0-5338 


3.9362 


0-7118 


4-9202 


0-8897 


79^ 


io>^ 


9 833 


1822 


9 665 


3645 


9498 


5467 


933o 


7289 


9163 


9112 


79^ 


i OK 


9825 


i865 


9649 


373o 


9474 


55 9 6 


9298 


7461 


9123 


9326 


79* 


11° 


9816 


1908 


9633 


38i6 


9449 


5724 


9265 


7632 


9081 


9540 


79° 


irK 


9808 


1961 


9616 


3902 


9424 


5853 


923i 


7804 


9039 


9755 


78K 


h& 


9799 


1994 


9598 


3987 


9 3 9 8 


5 9 8i 


9197 


7975 


8996 
8 9 52 


9968 


78* 


UK 


9790 

9781 


2o36 


9 58i 


4073 


9371 


6109 


9 i6p 


8146 


1-0182 


78* 


12° 


2079 


9 563 


4 1 58 


9344 


6237 


9126 


83i6 


8907 


0396 


78° 


12* 


9772 


2122 


9545 


4244 


9317 


6365 


9089 


8487 


8862 


0609 


77* 


12^ 


97 63 


2164 


9526 


4329 


9289 


6493 


9052 


8658 


881 5 


0822 


11H 


I2K 


o-97 53 


0-2207 


1.9507 


0-44U 


2-9260 


0-6621 


3-QOI4 
§ 97 5 


0-8828 


4-8767 


i-io35 


UK 


13° 


9744 


2250 


9487 


4499 


9231 


6740 


8998 


8719 


1248 


77° 


i3K 


9734 


2292 


9468 


4584 


9201 


6876 


8 9 35 


9168 


8669 


1460 


76K 


i3# 


9724 


2334 


9447 


4669 


9171 


7003 


88 9 5 


9 338 


8618 


1672 


7 ^X 


i3K 


97i3 


23 77 


9427 


4754 


9U0 


7.3. 


8854 


9 5o 7 


856 7 


1884 


76* 


14° 


97o3 


2419 


9406 


4838 


9109 


7258 


8812 


9677 


85i5 


2096 


7«o 


14* 


9692 


2462 


9385 


4923 


9077 


7385 


• 8769 


9846 


8462 


23o8 


7fJ5 


i4# 


9681 


25o4 


9363 


5oo8 


9044 


75 T T 


8726 


00! 5 


8407 


2619 


7fg 


14K 


9670 


2546 


934i 


5092 


901 1 
8978 


-63S 


8682 


0184 


8352 


2730 


7 5* 


15° 


9609 


2588 


9 3i 9 


5i 7 6 


7 7 65 


863 7 


o353 


8296 


2941 


75° 


Bear. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat, 


Bear. 




Dist. 1. 


Dist. 2. 


Dist. 3. 


Dist. 4. 


Dist. 5. 





LATITUDES AND DEPARTURES. 



65 



Bear. 


Dist. 6. 


Dist. 7. 


Dist. 8. 


Dist. 9. 


Dist. 10. 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. Dep. 


Bear. 


0% 


5.9999 


0-0262 


6-9999 


D-o3o5 


7.9999 


D-o349 


8-9999 


o-o393 


9.9999 


d-0436 


89% 


oH 


9998 


o524 


9997 


061 1 


9997 


0698 


9997 


0785 


9996 


0873 


8qH 


0% 


999D 


o 7 85 


9994 


0916 


9 99 3 


1047 


9992 


^H 


999 \ 


1 309 


89% 


1° 


9991 


1047 


9989 


1222 


9988 


1396 


9986 


1 57 1 


9980 


1745 


89° 


iU 


9986 


i3oqi 


9983 


1027 


9981 


1745 


9979 


1963 


9976 


2181 


88% 


i% 


9979 


i5 7 i 


9976 


i832 


9973 


2094 


9969 
9938 


2356 


9966 


2618 


88^ 


i% 


9972 


i832 


9967 


2i38 


9963 


2443 


2748 


9953 


3o54 


88% 


2° 


9963 


2094 


9957 


2443 


99 5 1 


2792 


9945 


3i4i 


99 3 9 


3490 


88° 


2% 


9954 


2356 


9946 


2748 


9938 


3i4i 


99 3 1 


3533 


9923 


3926 
4362 


87% 


2^ 


9943 


2617 


9933 


3o53 


9924 


3490 
3-3838 


9914 
8-9896 


3926 
o-43i8 


9905 


87^ 


2% 


5. 99 3i 


0-2879; 


6-9919 


3-3358 


7-9908 
9890 


9- 9 885 


0-4798 


87% 


3~ 


9918 


3 1 40: 


99°4 


3664 


4187 


9877 


4710 


9 863 


5234 


870 


3% 


99°4 


3402 


9887 


3 9 68 


987. 


4535 


9 855 


5l02 


9839 


566 9 


86% 


3H 


9888 


3663 


9869 


4273! 


9801 


4884 


9 832 


5494 


9 8i3 


6io5 


86% 


3% 


9872 


3924 


985o 


4578 


9829 


5232 


9801 


5886 


9786 


654o 


86% 


4° 


9804 


4i85 


9829 


4883 


980D 


558i 


.9781 


6278 


9756 


6976 


86° 


AM 


9 835 


4447 


9808 


5i88 


9780 


5 9 2 9 


97 53 


6670 


9725 


741 1 


85% 


AH 


9815 


4708 


9784 


5492 


9753 


6277 


9723 


7061 


9692 


7846 


85X 


4% 


9794 


4968 


9760 


5 797 


972D 


6625 


9691 


7453 


9657 


8281 


85% 


5° 


9772 


5229 


9734 


6101 


9696 


6972 


9 658 


7844 


9619 


8716 


85° 


5% 


5.9748 


• 5490 


6-9706 


3- 64o5 


7-9664 


0-7320 


8-9622 


0-8235 


9-9580 


o-9i5o 


sax 


5H 


9724 


5 7 5i 


9678 


6709 


9632 


7668 


9 586 


8626 


9540 


9585 


%ah 


5% 


9698 


601 1 


9648 


70i3 


9D97 


8oi5 


9 547 


9017 


9497 


1 -0019 


8aM 


6° 


9671 


6272 


9617 


7317 


9562 


8362 


9507 


9408 


9 452 


0453 


84° 


6% 


9643 


6532 


9 584 


7621 


9525 


8709 


9465 


9798 


9406 


0887 


83% 


6H 


9614 


6792I 


955o 


7924 


9486 


9056 


9421 


1-0188 


9357 


l320 


83% 


6% 


9384 


7062 


9 5i5 


8228 


9445 


94o3 


9376 


0578 


9 3 °7 


1754 


83% 


7° 


9 553 


7312 


9478 


853 1 


9404 


97 5o 


9 32 9 


0968 


9255 


2187 


83° 


lM 


9520 


7572 


944o 


8834 


9360 


1-0096 


9280 


i358 


9200 


2620 


82% 


1M 


9487 


7832 


9401 


9 l3 7 


93i6 


0442 


923o 


H47 


9i44 


3o53 


82% 


IX 


5-9452 


0-8091 
83 5o 


6-9361 


• 9440 


7-9269 


1-0788 


8-9178 


1-2137 


9-9087 


1-3485 


82% 


8° 


9416 


9 3l 9 


9742 


9221 


u34 


9124 


2526 


9027 


3917 


82° 


8% 


9379 


8610 


9276 


1 • 0044 


9172 


1479 


9069 


2914 


8965 


4349 


81% 


8H 


9341 


8869 


923i 


o347 


9121 


1825 


9011 


33o3 


8902 


4781 


81% 


8% 


93o2 


9127 


9i85 


0649 


9069 
0016 


2170 


8 9 53 


3691 


8836 


5212 


81% 


9° 


9261 


9 386 


9i38 


0950 


25i5 


8892 


4079 


8769 


5643 


81° 


9% 


9220 


9645 


9090 


1252 


8960 


285 9 


883o 


4467 


8700 


6074 


80% 


9H 


9177 


9903 


0040 


1 553 


8 9 o3 


3204 


8766 


4854 


8629 


65o5 


80% 


9% 


9 i33 


1-0161 


8989 


1 854 


8844 


3548 


8700 


524i 


8556 


6935 


80% 


10 J 


9088 


0419 


8 9 3 7 


2i55 


8 7 85 


38 9 2 


8633 


5628 


8481 


7365 


80° 


10% 


5-9042 


1-0677 


6-8883 


1-2456 


7.8723 


1-4235 


8-8564 


i- 601 5 


9-8404 


1.7794 


19 X 


io% 


8 99 5 


0934 


8828 


2756 


8660 


45 79 


8493 


6401 


8325 


8224 


79^ 


io% 


8947 


1191 


8772 


3o57 


85 9 6 


4922 


8421 


6787 


8245 


8652 


19M 


110 


8898 


1449 


87U 


335 7 


853o 


5265 


8346 


71-73 


8i63 


9081 


79° 


n% 


8847 


i 7 o5 


8655 


3656 


8463 


5607 


8271 


7558 


8079 


9509 


78% 


uH 


8 79 5 


1962 


85 9 5 
8533 


3 9 56 


83 9 4 


5949 


8i 9 3 


7943 


7992 


99 3 7 


7 «$ 


ii% 


8743 


2219 


4255 


8324 


6291 


8114 


8328 


79 o5 
7815 


2-o364 


78% 


12° 


8689 


2475 


8470 


4554 


8252 


6633 


8o33 


8712 


0791 


78° 


123^ 


8634 


2731 


8406 


4852 


8178 


6974 
73i5 


79 5i 
7867 


9096 


7723 


1218 


77^ 


12% 


85 7 8 


2986 


834i 


5i5i 


8104 


9480 


763o 


1644 


llH 


12% 


5-8521 


1-3242 


6-8274 


1 • 5449 


7-8027 


1-7656 


8-7781 


i- 9 863 


9-7534 


2-2070 


77^ 


13° 


8462 


3497 
3752 


8206 


5747 


7o5o 


7996 


7693 


2-0246 


7437 


2495 


77° 


i3% 


84o3 


8137 


6044 


7870 


8336 


7604 


0628 


7338 


2920 


76% 


i3X 


8342 


4007 


8066 


634i 


779° 


8676 


75i3 


1010 


7237 


3345 


76% 


i3% 


8281 


4261 


7994 


6638 


7707 


9015 


7421 


1392 


7i34 


3769 


76% 


14° 


821E 


45i5 


7921 


6 9 35 


7624 


9354 


7327 


1773 


7o3o 


4192 


76° 


lAM 


8i5^ 


\ 4769 


7846 


723i 


7538 


9692 


7231 


2 1 54 


6923 


461 5 


75% 


UX 


808c 


) 5o23 


7770 


7527 


7452 


2-Oo3o 


7 i33 


2534 


68i5 


5o38 


75% 


14% 


802; 


5 27 6 


7693 


7822 


7364 


o368 


7034 


2914 


6705 


6460 


75% 


15° 


79 5( 


) 5529 


7615 


8117 


7274 


0706 


6 9 33 


3294 


65 9 3 


5882 


75° 
Bear. 


Bear. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


DiST. 6. 


Dist. 1. 


Dist. 8. 


Dist. 9. 


Dist. 10. 





19 



66 



LATITUDES AND DEPARTURES. 





Dist. 1. 


Dist. 2. 


Dist. 3. 


Dist. 4. 


Dist. 5. 




Bear 












Bear. 

74K 




Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


\5% 


0-9648 


o-263o 


1 -9296 


o-526i 


2-8944 


0-7891 


3-85 9 i 


1 -o52i 


4-823q 


i-3i52 


i5% 


g636 


2672 


9273 


5345 


8909 

8874 


8017 


8545 


0690 


8182 


3362 


74% 


1 5% 


9625 


27U 


9249 


5429 


8i43 


8408 


o858 


8i23 


3572 


74X 


16° 


9613 


2756 


9220 


55i3 


8838 


8269 


845c 


1025 


8o63 


3782 


740 


16% 


9600 


2798 


9201 


55 97 


8801 


83 9 5 


8402 


1193 


8002 


3991 


l3% 


i(>X 


9 588 


2840 


9176 


568o 


8 7 65 


8520 


8353 


]36i 


794i 
787c. 
7 8i5 


4201 


73% 


ftx 


9 5 7 6 


288' 


9i5i 


5764 


8727 


8646 


83o3 


i528 


44io 


j3% 


iyo 


9563 


2924 


9126 


5847 


8689 


8771 


8252 


1695 


4619 


73o 


m 


955o 


2965 


9100 


5931 


865 1 


8896 


8201 


1862 


773i 


4827 


72% 


nx 


9 53 7 


3007 


9074 


6014 


8612 


9021 


8149 


2028 


7686 


5o35 


72^ 


i7M 


0-9524 


o-3o49 


1-9048 


0-6097 


2-85 7 2 


0-9146 


3-8096 


1-2195 


4-7620 


1-5243 


72% 


18° 


95n 


3090 


9021 


6180 


8532 


9271 


8042 


236i 


755o 


545 1 


72° 


18% 


9497 


3i32 


8994 


6263 


8491 


93 9 5 


7988 


2527 


7483 


5658 


71^ 


18X 


9 483 


3i 7 3 


8966 


6346 


8400 


9519 


7933 


2692 


74i6 


5865 


V% 


18% 


9469 


3214 


8 9 3 9 


6429 


8408 


9643 


7877 


2858 


7347 


6072 


-]i% 


19° 


9455 


3256 


8910 
8882 


65n 


8366 


9767 


7821 


3023 


7276 


6278 


71° 


19^ 


9441 


32 97 


6594 


8323 


989, 


7764 


3i88 


7204 


6485 


70X 


19X 


9426 


3338 


8853 


6676 


8279 


1 -0014 


7706 


3352 


7i32 


6690 


70^ 


19^ 


9412 


3379 


8824 


6 7 58 


8235 


oi38 


7647 


35i 7 


7o5 ? 


6896 


-]o% 


20° 


9 3 97 


3420 


8794 


6840 


8191 


0261 


7588 


368 1 


6985 


7101 


700 


20% 


0-9382 


o-346i 


1-8764 


0-6922 


2-8146 


i-o384 


3.7528 


1-3845 


4-6910 


1 -7306 


69% 


20J* 


9 36 7 


35o2 


8 7 33 


7004 


8100 


o5o6 


7467 


4008 


6834 


75io 


69X 


20% 


935i 


3543 


8 7 o3 


7086 


8o54 


0629 


74o5 


4172 


6 7 5 7 


77 i5 


69K 


21° 


9336 


3584 


8672 


7167 


8007 


0751 


7343 


4335 


6679 


79,8 


69° 


21% 


9320 


3624 


8640 


7249 


7960 


0873 


7280 


4498 


6600 


8122 


68% 


l\% 


93o4 


3665 


8608 


733o 


79»3 


0995 


7217 


4660 


652i 


8325 


68% 


i\X 


9288 


3706 


8576 


74i 1 


7864 


1117 


7i52 


4822 


6440 


8528 


68^ 


zz° 


9272 


3746 


8544 


7492 


7816 


1238 


7087 


4984 


635 9 


8730 


6S° 


22% 


9255 


3786 


85n 


7 5 7 3 


7766 


1359 


7022 


5i46 


6277 


8 9 3 2 


67% 


22% 


9239 


382 7 


8478 


7654 


7716 


1481 


6 9 55 


53o7 


6194 


9i34 


67^ 


22% 


0-9222 


o- 386 7 


1-8444 


0-7734 


2.7666 


1-1601 


3-6888 


1-5468 


4-6no 


1 -9336 


67M 


23° 


9205 


3907 


8410 


78i5 


76i5 


1722 


6820 


5629 


6o25 


9537 


67° 


23% 


9188 


3947 


8376 


7895 


7564 


1842 


6752 


5790 


5940 


9737 


66% 


2ZX 


9171 


3987 


834i 


7973 


75i2 


1962 


6682 


5950 


5853 


99 3 7 


66% 


2VA 


oi53 


4027 


83o6 


8o55 


7459 


2082 


6612 


6110 


5766 


2-0137 


66% 


24° 


9 i35 


4067 


8271 


8i35 


74o6 


2202 


6542 


6269 


56 77 


o337 


66° 


24^ 


9118 


4107 


8235 


8214 


7353 


2322 


6470 


6429 


5588 


o536 


65% 


24^ 


9100 


4i47 


8199 


8294 


7299 


2441 


63 9 8 


6588 


5498 


0735 


65% 


24% 


9081 


4187 


8i63 


8373 


7244 


256o 


6326 


6746 


5407 


0933 


65% 


25° 


9063 


4226 


8126 


8452 


7189 


2679 


6252 


6905 


53i5 


ii3i 


65° 


25% 


0-9045 


3-4266 


1-8089 


o-853 1 


2-7134 


1 • 2797 


3-6178 


1 .7063 


4-5223 


2-1328 


64% 


25% 


9026 


43o5 


8o52 


8610 


7078 


2915 


6ro3 


7220 


5i 29 

5o35 


1 526 


64% 


25% 


9007 
8988 


4344 


8014 


8689 


7021 


3o33 


6028 


7 3 7 8 


1722 


64% 


26° 


4384 


7976 


8767 


6964 


3i5i 


5952 

58 7 5 


7535 


4940 


1919 


64° 


26% 


8969 


4423 


7937 
7899 


8846 


6906 


3269 


7692 


4844 


21 14 


63% 


26% 


8949 


4462 


8924 


6848 


3386 


5797 


7848 


4747 


23lO 


63% 


26% 


8 9 3o 


45oi 


7860 


9002 


6789 


35o3 


5719 


8004 


4649 


25o5 


63 % 


27° 


8910 


454o 


7820 


9080 


6730 


3620 


5640 


8160 


455o 


2700 


63° 


27% 


8890 


4579 


778o 


9157 


6671 


3 7 36 


556i 


83 1 5 


445 1 


2894 
3087 


62% 


27X 


8870 


4617 


7740 


9235 


6610 


3852 


5480 


8470 


435 1 


62% 


27% 


o-885o 


5-4656 


1 -7700 


o-93i2 


2-655o 


1-3968 


3-54oo 


1-8625 


4-4249 


2-328i 


62% 


28° 


8829 


469D 


7659 


9389 


6488 


4084 


53i8 


8779 
8 9 33 


4147 


3474 


62° 


28% 


8809 
8788 


4733 


7618 


9466 


6427 


4200 


5236 


4045 


3666 


61% 


28^ 


4772 


7 5 7 6 


9543 


6365 


43i5 


5i53 


9086 


3941 


3858 


6i>* 


2%% 


8767 


4810 


7535 


9620 


63o2 


443o 


5069 


9240 


3836 


4049 


61^ 


29° 


8746 


4848 


7492 


9696 


6239 
6i 7 5 


4544 


4985 


9 3 9 2 


373i 


4240 


61° 


29% 


8725 


4886 


7430 


9772 


465g 


4900 


9545 


3625 


443 1 


60% 


29^ 


8704 


4924 


7407 


9848 


61 1 1 


4773 


4814 


9697 


35i8 


4621 


60% 


29X 


8682 


4962 


7364 


9924 


6046 


4886 


4728 


9849 


34io 


481 1 


60^ 


30° 


8660 


5ooo 


7321 


I -0000 


5 9 8i 


5ooo 


4641 


J -0000 


33oi 


5ooo 


60° 

3ear. 


Bear. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. Lat 


Dep. 


Lat. 


DlST. 1. 


Dist. 2. 


Dist. 3. 


Dist. 4. 


Dist. 5. 





LATITUDES AND DEPARTURES. 



67 



Bear. 


Dist. 6. 


Dist. 7. 


Dist. 8. 


Dist. 9. 


Dist. 10. 


Bear. 

I 


Lat 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat Dep. 


1 l5x 

i5% 

i k% 
16 y 

16% 
ibX 

1 6% 
11° 
17* 
17* 
17% 
18° 
18% 

18% 
19 J 

w* 

19* 
«9& 
20 J 


'5^7887 
7818 
7747 
7676 
76o3 
7 52 9 
7454 
7 3 7 8 
73oi 
7223 

5.7144 
7063 
6982 
6899 
6816 
6731 
6645 
6558 
6471 
6382 


1 -5 7 82 
6o34 
6286 
6538 
6790 
7041 
7292 
7042 
7792 
8042 

1-8292 
854i 
8790 
9038 
9286 
9334 
9781 

2-0028 
0275 

0521 


6.7535 
7454 
7372 
7288 

T203 
7117 

7o3o 
6941 
685 1 
6760 
6-6668 
6574 
6479 
6383 
6285 
6186 
6086 
5o85 
5882 
5 77 8 


I- 8412 
8707 
9001 
9295 
9588 
9881 

2-0174 
0466 
o 7 58 
1049 

2-i34r 
i63i 
1921 
2211 
25oi 

2790 
3078 

3366 
3654 
394i 


7-7183 
7090 
6996 
6901 
6804 
6706 
6606 
65o4 
6402 
6297 

7-6192 
6o83 
5976 
5866 
5754 
564i 
5527 
54ii 
3294 
5175 


2-1042 
1379 
1713 

2031 

2386 
2721 
3o56 
3390 
3723 
4o56 
2- 4389 
4721 
5o53 
5384 
57i5 
6o45 
6375 
6705 
7o33 
7 362 


8-683i 

6727 
6621 
65i4 
6404 
6294 
6181 
6067 
5952 
5835 
8.5716 
55 9 5 
5473 
5349 

5224 

5097 
4968 
4838 
4706 
4572 


2.3673 
4o5i 
443o 
4807 
5i85 
556i 
5 9 38 
63i3 
6689 
7064 

2-7438 
7812 
8i85 
8557 
8o3o 
9301 
9672 

3-oo43 
o4i3 
0782 


9.6479 
6363 
6246 
6126 
6oo5 
5882 
5 7 5 7 
563o 
55o2 
53 7 2 

9-524o 
5io6 
4970 
4832 
46o3 

4532 

4409 

4264 
4118 
3969 


|2-63o3 
6724 
7144 
7564 
79 83 
8402 
8820 
9237 
9 654 
3-0071 
3.0486 
0902 
i3i6 
1730 
2144 
2557 

2969 
338i 

3792 
4202 


74% 

74% 

74% 

74 

7 3% 

73*r 

73X 
73° 
72% 
72# 
72% 
72° 
li% 
7i# 

im 

71 

70% 
70% 
7°* 
70° 


20% 
20% 
20% 
21° 
21% 
21% 

*»X 

22 : % ! 

22% 
22% ! 

23% 
23% 

23^ 1 

ZA- 1 

2AH ! 
24^ i 
24% | 
25° 


5-6291 
6200 
6*o8 
601 5 

5o20 
5823 

5729 

563 1 
5532 
5433 
5-5332 
523o 
3127 
5o24 
4919 
48i3 
4706 
4598 
4489 
4378 


2-0767 

IOI2 

1257 
l502 

1746 
1990 

2233 

2476J 
2719; 

2961 ' 

2-3203 

3444 

3685! 
3 9 25 
4i65| 

44o4 
4643; 

4882 

5l20 

5357 


6-56 7 3 
.5567 
5459 
535i 
524i 
5129 
5oi7 
4903 
4788 
4672 

6-4554 
4435 
43 1 5 
4194 
4072 
3 9 48 
3823 
3697 
3570 
3442 


2-4228 
45i5 
4800 
5o86 
5371 
5655 
5939 
6222 
65o5 
6788 

2.7070 
735i 
7632 
7912 
8192 
8472 
8750 
9029 
93o6 
9583 


7«5o55 
4934 
4811 
4686 
456 1 
4433 
43o5 

4n5 
4o43 
3910 
7.3776 
364o 
35o3 
3365 

3225 

3o84 
2941 

2797 
2631 
25o5 


2-7689 
8017 
8343 
8669 
8995 
9320 
9645 
9969 

3-0292 
o6i5 

3-0937 
1258 
i58o 
1900 
2220 
2539 
2858 
3n5 
3493 
38o 9 


8.4437 
43oo 
4162 
4022 
388i 
3738 
3593 
3447 
3299 
3 149 

8-2998 
2845 
2691 
2535 
23 7 8 
2219 
2059 
1897 
i 7 33 
1 568 


3 • 1 1 5i 

1519 

1886 

2253 

2619 

2 9 85j 
335o| 
37i5i 
4078 
4442 
3.4804 
5i66 
5527 
5887 
6247 
6606 
6965 
7322 

7 6 79 ! 
8o36j 


9.3819 
3667 
35i4 
3358 

3201 

3o42 
2881 
2718 
2554 
2388 
9-2220 
2o5o 

1879 

1706 
i53i 
1355 
1 1 76 
0996 
0814 
o63i 


3.4612 

5o21 

5429 
583 7 
6244 
665o 
7o56 
746i 
7865 
8268 

3.8671 
9073 
9474 
9875 

4-0275 
0674 
1072 
1469 
1866 
2262 


69% 
69X 
69^ 
69* 
68% 
68% 
68% 
68° 
67% 
67X 
67K 
G7° 
66% 
66% 
66% 
6Q° 
65% 
65% 
65% 
65° 


25¥ 1 

25% | 
25% 

26% i 
26% i 

26% 

27* ! 

27X 
27X j 

28% ! 
28^ ! 

28% ! 

2 9 % i 

29X 
29% 

Bear. 


5-4267 
4i55 
4042 
3928 
38i2 

It, 

3460 
3341 

3221 

5- 3099 
2977 
2853 
2729 
2604 
2471 

23JO 
2221 
2092 
I962 


2-5594! 

583 1 
6067 1 
63o2 
653 7 1 
6772! 
7006; 
723 9 
7472; 

7703; 
2 -7937: 

8168 
S3 99 
863o 
8859 
9089 
9 3i 7 
9545; 
, 9773; 

3 -OOOO; 


6-33i2 
3i8i 
3 049 
2916 
27S1 
2645 
2509 
2370 

223 I 

2091 

6-1949 

1806 

1662 

1517 
1 37 1 

1223 
I0 7 5 
0925 

0774 
0622 


2-9860 
3-oi36 
041 1 
0686 
0960 
1234 
007 
1779 
2o5i 

2322 
3-2593 

2863 
3i32 
34oi 

3669 

3937! 
42o3 

4470 
4735 
5ooo 


7-2356 
2207 
2o56 
1904 
ipo 
095 
1438 
1281 
1121 
0961 

7-0799 
o636 
0471 
o3o5 
or38 

6-9970 
9800 
9628 
9456 
9282 


3-4i25 
444i 

4756 
5070 
5383 
56 9 6 
6008 
63 19, 
663 oj 
6940 
3-7249 
7558 
7866 
8173 

8479 
8785 
9090 
9 3 9 4 
9697 
4-0000 


8-i4oi 
1233 
io63 
0891 
0719 
o544 
o368 
0191 
0012 
7-9831 
7.9649 
9465 
9280 
Q094 

8525! 
8332 
8i38 
7942 


3-839i; 
8746! 
9100 
9453 
9806! 

4-oi58; 
0509 
0859 

I209 ! 

1 557 
4-1905! 

2252 

2599; 

2944 
3289! 

3633! 
3976! 
43i8 
465 9 ! 
5ooo 


9-0446 
0259 
0070 

8.9879 
9687 
9493 
9298 
9101 
8902 
8701 

8-8499 
8295 
8089 
7882 
7673 
7462 
725o 
7o36 
6820 
66o3 


4.2657 
3o5i 
3445 
333 7 
4229 
4620 
5oio 
5399 
5787 
6i 7 5 

4-656i 
6947 
7 332 
7716 
8099 
8481 
8862 
9242 
9622! 

5-oooo 


64% 
64% 
64% 
64° 
63% 
63% 
63% 
63° 
62% 
62% 
62% 
62° 
61% 
61% 
61% 
61° 
61% 
61% 
61% 
00° 


Dep. 1 Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. j Lat. 


Bear. 


DlST. 6. 


Dist. 7. 


Dist. 8. 


Dist. 9. 


Dist. 10. 












IS 


1 













68 



LATITUDES AND DEPARTURES. 



Bear. 


Dist. 1. 


Dist. 2. 


Dist. 3. 


Dist. 4. 


Dist. 5. 








1 














Bear. 




Lat. 


Dep. 


Lat. Dep. 


Lat. 


Dep. 


Lat. 


Dep. 


Lat 


Dep. 




3o% 


0-8638 


o-5o38 


1-7277 


1-0075 


2.59i5 


i«5u3 


3-4553 


2«Ol5l 


4-3i92 


2.5189 


5g% 


3o% 


8616 


D075 


7233 


oi5i 


5849 


5226 


4465 


0302 


3o8i 


53 7 7 


$9% 


3o% 


8094 


5n3 


7188 


0226 


5782 


533 9 


4376 


0452 


2970 


5565 


£ 1 

5 9>4 ' 


31° 


8672 


5i5o 


7143 


o3oi 


5715 


545 1 


4287 


0602 


2858 


5 7 5 2 


59° j 


3i% 


8049 


5i88 


7098 
7o53 


o3p 
0400 


5647 


'5563 


4196 


0751 


2746 


5939 

6l2D 


58% ; 


3i% 


8026 


5225 


55 79 


56 7 5 


4106 


0900 


2632 


58% : 


3i% 


85o4 


5262 


7007 


o524 


55n 


5786 
58 9 8 


4014 


1049 


25i8 


63n 


58% 


32° 


8480 


5299 


6961 


o5g8 


544i 


3922 
3829 


1 197 
i345 


2402 


6496 


58° ■ 


32% 


8457 


5336 


6915 


0672 


5372 


6008 


2286 


6681 


57% 


32% 


8434 


53 7 3 


6868 


0746 


53o2 


6119 


3 7 36 


1492 


2170 


6865 


$7% 


32% 


0-8410 


0-5410 


i- 6821 


1-0819 
o8 9 3 


2-523l 


1-6229 


3.3642 


2-1639 


4-2o52 


2-7049 


57% ! 


33 J 


838 7 


5446 


6773 


5i6o 


633 9 


3547 


1786 


1934 


7232 


57o 


33% 


8363 


5483 


6726 


0966 


6089 


6449 


345 1 


1932 


1814 


74i 5 


56% 


33% 


833o 


55i9 


6678 


1039 


5017 


6558 


3355 


2077 


1694 


7^97 


563* 


33% 


83iD 


5556 


6629 


1111 


4Q44 


6667 


3259 


2223 


i5 7 3 


7779 


56% 


34° 


8290 


5592 


658i 


1184 


4871 


6776 


3i62 


2368 


1452 


7960 


58° 


34% 


8266 


5628 


6532 


1256 


4798 


6884 


3o64 


25l2 


1329 


8140 


55% 


34% 


8241 


5664 


6483 


i328 


4724 


6992 


2965 
2866 


2656 


1206 


8320 


55% 


34% 


8216 


5700 


6433 


1400 


4649 


7100 


2800 


1082 


85oo 


55% 


35° 


8192 


5736 


6383 


1472 


4375 


7207 


2766 


2943 


0958 


8679 


55° 
54% 


35% 


0-8166 


0.5771 
■ 58o 7 


1-6333 


i.i543 


2 • 4499 


i-73i4 


3 • 2666 


2-3o86 


4-o832 


2-8857 


35% 


8141 


6282 


1614 


4423 


7421 


2565 


3228 


0706 
0579 


9o35 


54% 


35% 


8116 


5842 


623i 


i685 


4347 


7 52 7 


2463 


3370 


9212 


54X 


36° 


8090 


58 7 8 


6180 


1756 


4-271 


7634 


236i 


35n 


045 1 


9389 


54° 


36% 


8064 


5913 


6129 


1S26 


4193 


77 3 9 


2 258 


3652 


0322 


9565 


53% 


36% 


8039 
8oi3 


5 9 48 


6077 


1896 


4n6 


7845 


2154 


3793 


OI93 


974i 


53% 


36% 


5 9 83 


6025 


1966 


4o38 


7 9 5o 


2o5o 


3 9 33 


oo63 


9916 


53% 


37° 


7986 


6018 


D973 


2o36 


39D9 


8o54 


1945 


4073 


3-9932 


3-0091 


53° 


3 7 % 


7960 


6o53 


5920 


2106 


388o 


8i5g 


1840 


4212 


9800 


0265 


52% 


37% 


7934 


6088 


5867 


2175 


38oi 


8263 


1734 


43 5o 


9668 


0438 


5a& 


37% 


0.7907 


0-6l22 


i-58i4 


1-2244 


2-3721 


i-836 7 


3-1628 


2-4489 


3-9534 


3-o6n 


52% 


38° 


7880 


607 


5760 


23i3 


364o 


8470 


I 520 


4626 


9401 


0783 


52° 


38% 


7853 


619I 


5706 


2382 


356o 


8573 


I4i3 


4764 


9266 


0955 


5i% 


38% 


7826 


6225 


5652 


245o 


3478 


8675 


i3o4 


4901 


9i3o 


1126 


5i% 


38% 


7799 


6259 


55 9 8 


25i8 


3397 


8778 


1195 


5o3 7 


8994 

8857 


1296 


5i% 


39° 


7771 


6293 


5543 


2586 


33 1 4 


8880 


1086 


5i 7 3 


1466 


51° 


3 9 % 


7744 


b327 


54b8 


2654 


3232 


8981 


0976 


53o8 


8720 


i635 


5o% 


3g% 


7716 


636i 


5432 


2722 


3i4g 


9082 


0865 


5443 


858i 


1804 


5o% 


3 9 % 


1 768b 


63g4 


5377 


2789 


3o65 


9183 


0754 


5578 


8442 


1972 


5o% 


40° 


7660 


6428 


5321 


2856 


2981 


9284 
1 • 9384 


0642 


0712 


83o2 
3-8162 


2139 
3-23o6 


5©^ 


4o% 


0-7632 


0-6461 


j 1. 5265 


1.293* 


2-2897 


3- 0529 


2-5845 


49X 


40% 


7604 


6494 


5 2 08 


2989 


2bl2 


9483 


0416 


5 97 8 


8020 


2472 


49% 


40% 


7 5 7 6 


652b 


5i5i 


3o55 


2727 


9583 


o3o3 


6110 


787b 


2638 


49% 


41° 


7547 


656i 


5094 


3l2I 


264 1 


9682 


0188 


6242 


7735 


?.8o3 


490 


41K 


75i8 


6593 


D037 


3187 


2555 


9780 


0074 


6374 


7392 


2967 


48% 


41% 


7490 


6626 


4979 


3252 


2469 


9879 


2- 99 58 


65o5 


7448 


3i3i 


48% 


4i% 


7461 


6609 


4921 


33i8 


2382 


9976 


9842 


6635 


73o3 


3294 


48% 


42° 


743i 


6691 


4863 


3383 


229.4 


2-0074 


9726 


6765 


7i5 7 


345 7 


48° 


42% 


7402 


6724 


4804 


3447 


2207 


0171 


9609 


6895 


7011 


36i8 


47% 


42% 


7 3 7 3 


6756 


4746 


35i2 


,21 lb 


026b 


9491 


7024 


6b64 


3780 


47% 


42% 


0.7340 


0-6788 


1-4686 


1.3576 


2-2030 


2-o364 


2-9373 


2-7l52 


3-6716 


3 • 3940 


47% 


43° 


73i<; 


6820 


4627 


3640 


1 94 1 


0460 


9204 


72«o 


6568 


4100 


47^ 


43% 


728^ 


6852 


4567 


3704 


. i85i 


o555 


9i35 


7407 


6419 


425q 


46% 


43^ 


72DZ 


. 6884 


4007 


3 7 6 7 ; 1761 


o65i 


901D 


7534 


6269 


4418 


46% 


43% 


722/ 


6915 


1 4447 


383o! 1671 


07451 8895 


7661 


6nb 


4576 


46% 


44° 


719; 


5 6947 


1 4387 


3893 


i58c 


0840J 877/ 
J 0934] 8652 


7786 


0967 


4733 


46 J 


44% 


716, 


5 697 b 


4326 


3g5t 


i48§ 


79'2 


5b 1 5 


4890 


45% 


44% 


7i3v 


} 7009 


, 426: 


40 lb 


i3gr 


1027J 853o 


8o36 


5663 


5o45 


45% 


44 X 


710 


2 704c 


! 4204 


4.0S0 


i3or 


• 1120IJ 8407 


8161 


5509 


5201 


45% 


45° 


707 


1 707 1 


4142 


4I42J 12l3 


I2l3|! 8284 


8284 


5355 


5355 


450 




Dep. 


Lat. 


Dep. 


Lat. 


| Dep. 


Lat. || Dep. 


Lat. 


Dep. 


Lat. 




Bear 














Bear. 


DlST. 1. 


Dist. 2. 


| Dist. 3. 


Dist. 4. 


i Dist. 5. 





LATITUDES AND DEPARTURES. 





Dist. 6. 


Dist. 7. 


Dist. 8. 


Dist. 9. 


Dist. 10. 




Bear. 






















Bear. 


3o% 


Lat 


Dep. 


Lat. 


Dep. j 


Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. 




5-i83o 


3-0226J 


6-0468 


3-52641 


6-oi074-o3o2 
8930! o6o3 


7-7745 


4- 534o | 


8-6384 


5-o377] 


39% 


3o% 


1698 


0452; 


o3i4 


55 2 8j 


7547 


5678I 


6i63 


07541 


59% 


3o% 


1 564 


0678I 


oi58 


5791 1 
6o53' 


8 7 53 o 9 o3| 


7347 


6016! 


5941 


1129 


5 9% 


31 u 

3i% 


i43o 


0902 


0002 


85t3! i2o3' 


7145 


6353! 


5717 


i5o4! 


59 


1295 


1126 


5-9844 


63i4| 


8393! 1502 


6942 


6690! 


5491 


1877I 


58% 


3i% 


n58 


i35o 


968D 


6575 


8211! i8ooj 


6738 


7°2 5 


5264 


2250 


58% 


3i% 


1021 


i5 7 3| 


9525 


6835 


8028I 20971 


6532 


735o 
i6 9 3 
8025 


5o35 


2621 


58k 


32- 


o883 


1795 


9 363 


7094 


7844I 2394 


6324 


48o5 


2992 


58° 


32% 


0744 


2017 


9201 


7353 


76D8 


2689 


6116 


4573 


336i 


5 7 % 


32% 


o6o3 


2238 


9037 
5.8873 


761 1 


747i 


2984 


5905 


8357 


433g 


3730; 


57% 


32% 


5-0462 


3-2458 


J- 7868 
8i25' 


6- 7 283 


4-3278 


7-56o4 4-8688j 
5480 9018 


8-4J04 


5-4097; 


57% 


33^ 


o3ao 


2678 


8707 


7094 


3571 


3867 


4464! 


57° 


33% 


0177 


2898 


854o 


833i 


6903 


3863 


5266 9346, 


3629 


4829! 


56% 


33% 


oo33 


3u6 


8372 


8636! 


67 1 1 


4i55 


5o5o 


■ 9074 


338 9 


5i 9 4; 


56% 


33% 


4-9888 


3334 


82o3 


8890 


65i8 


4446 


4832 


5-oooi 


3i47 


5557! 


56% 


34 J 


9742 


3552 


8o33 


.9144 


6323 


4735 


46i3 


0327 


2904 


5919 


56 


34% 


9595 


3 A 8 


7861 


9396 


6127 


5o24 


43g3 


o652 


2659 
24i3 


6280; 


55% 


34% 


9448 


3984 


7689 


9648 


5930 


53i2 


4171 


0977 


6641! 


55% 


34% 


9299 


4200 


75i5 


9900 


5 7 32 


56oo 


3 9 48 


i3oo 


2i65 


7000] 


55% 


35° 


9149 


44i5 


734i 


i«oi5o 


5532 


5886 


3724 


1622 


igi5 

8-1664 


7358 

5.7715 


55° 

54% 


35% 


4.8998 
8847 


3.4629 


5-7165 


4-0400 


6-533i 


4-6172 


7-3498 


5-1943 


35% 


4842 


6988 
6810 


0649 


5129 


6456 


3270 


2263 


1412 


8070 


54% 


35% 


8694 


5o55 


0897 


4926 


6740 


3042 


2582 


n5 7 


842 5 ( 


54% 


36 3 


854i 


5267 


663 1 


1 145 


4721 


7023 


2812 


2901 


0902 8779' 


54° 


36% 


838 7 


5479 


645 1 


1392 


45i6 


73o5 


258o 


3218 


0644 


9i3ij 


53X 


36% 


8a3i 


5689 


6270 


1638 


4309 


7586 


2347 

2Il3 


3534 


o386 


9482; 


53% 


36% 


8075 


58 99 


6088 


1 883 


4100 


7866 
8i45 


3849 < 


0125 


9 832 ! 


53% 


37o 


7918 


6109 
63i8 


5904 


2127 


38 9 i 


1877 


41 63,1 


7-9864 


6-0182! 


53° 


37K 


7760 


5720 


2371 


368o 


8424 


1640 


4476 


9600 


0529J 


52% 


37% 


7601 


6526 


5535 


26i3 


3468 


8701 


1402 


4789 


9335 


0876 


52% 


37% 


4-7441 


3-6733 


5.5348 


4-2855 


6-3255 


4-8977 


7.1162 


5-5ioo 


7-9o69:6-l222' 


52% 


38° 


7281 


6940 


5i6i 


3096 
3337 


3o4i 


9253 


0921 


54io 


880 1 j 1 566 


52° 


38% 


7119 
6 9 56 


7U6 


4972 


2825 


9 528 


0679 


5718 1 


8532; 1909 


5i% 


38# 


735i 


4783 


3576 


2609 


9801 


0435 


6026 


8261 22DI 


5i% 


38% 


6793 


7555 


4592 


38i5 


2391 


5.0074 


0190 


6333 


7988 2592 


5i% 


39 J 


6629 


77 5 9 


4400 


4o52 


2172 


o346 


6 • 9943 


663 9 


77i5| 2g32 


51° 


3 9 % 


6464 


7962 
8i65 


4207 


4289 


1961 


0616 


9695 


6943 


7439 


3271 


5o% 


3g% 


6297 
6i3i 


4014 


4525 


1730 


0886 


9446 


7247 


7162 


36o8 


5o% 


39X 


8366 


38i 9 


4761 


1 507 


u55 


0196 


7530 


6884 


3944 


5o% 


40° 


5 9 63 


8567 


3623 


4995 


1284 


1423 


8944I 785i 


6604 


4279 


50° 


4o% 


4-5794 


3.8767 


5.3426 


4-5229 


6-ioog 


5-1690 


6-8691 5-8i5i 


7-6323«6-46i2 


49^ 
49% 
49% 
490 


4o% 


5624 


8967 


3228 


546i 


o832 


1906 


8437 845o 


6041 4945 


40% 


5454 


9166 


3o3o 


56 9 3 


o6o5 


2221 


8181 


8748 


5756J 5276 


41« 


5283 


9364 


283o 


5924 


0377 


2485 


7924 


9045 


54.71 1 56o6 


41% 


5no 


9561 


2629 


6i54 


0147 


2748 


7666 


9341 


5 1 84I 5 9 35 


48% 


4i% 


4937 


97 5 7 


2427 


6383 


5-9916 


3oio 


7406 


9636 


4896 6262 
4606! 6588 


48% 


4i% 


4763 


9953 


2224 


6612 


9685 


3271 


7143 


9929 


48% 


42o 


45894-0148 


2020 


683 9 


9452 


353o 


68836-0222 


43m 


6913 


48° 


42% 


44i3 o342 


i8i5 


7066 


9217 


3789 


6620! o5i3 


4022 


7237 


47% 


42% 


4237 


0333 


1609 


7291 


1 8982 


4047 


6355 o8o3 


3728 


7539 


47% 


42% 


4- 4059 


4-0728 


5-14034.7516 


6-8746 


5-43o4 


6-60896-1092 


7-34326-7880 


47% 


43° 


388i 


0920 


1 195 
0986 


7740 


85o8 


456o 


5822J i38o 


3i35 


8200 


47° 


43% 


3702 


1III 


7963 
8i85 


8270 


481 5 


5553 1666 


283-] 


85i8 


46% 


43% 


3522 


i3oi 


0776 


8o3o 


5o68 


5284 1952 


253- 


8835 


46% 


43% 


3342 


U91 


o565 


8406 


7789 


532i 


5oi3 2236 


223d 


9i5i 


46% 


44° 


3i6o 1680 


o354 


8626 


7547 


55 7 3 


474i 2519 


i 9 34 


9466 


4 6° 


44% 


2978' 1867 


0141 


8845 


73o<, 


5823 


44671 2801 


i63c 


9779 II 43 % 


44% 


2795I 2o55 


4-9928 


9064 


706c 


6073 


4i93i 3o82 


1323,7-0091 45^ 


44% 


2611J 2241 


9713 


9281 


68i< 


632i 


3917 336i 


1019 0401!! tfx 


45° 


2426 ( 2426; g 4g^ 

1 


9497 


656c 


1 6569 


3640 3640 


1 071 1 j 071 1 


1 450 


: Dep. i Lat. 


Dep. 


Lat. 


Dep. 


Lat. 


Dep. | Lat. 


\ Dep. j Lat. 




Bear. 












| ' 


Bear. 




| Dist. 6. 


Dist. 7. 


Dist. 8. 


Dist. 9. 


1 Dist. 10. 

1 



TABLE IV. 



CONTAINING 



CORRECTIONS OF THE MIDDLE LATITUDE 



OE, 



THE QUANTITIES TO BE ADDED TO THE MIDDLE LATITUDE, 

IN ORDER TO FIND THE LATITUDE IN WHICH THE 

DISTANCE BETWEEN THE MERIDIAN SAILED 

FROM AND THE ONE ARRIVED AT IS 

EQUAL TO THE DEPARTURE. 



72 



COKKECTIONS OF MIDDLE LATITUDE. 



SB 

i 

a >S 



15 


DIFFERENCE OF LATITUDE. 


1° 




2° 
I 


3° 

2 


4° 
3 


5° 
5 


6° 

I 


7° 


8° 


9° 


10° 


11° 


12° 


13° 


14° 


15° 


16° 


17° 


18° 


19° 


20° 


9 


12 


i5 


18 


22 


26 


3i 


36 


4i 


47 


52 


5 9 


Cj 


72 


16 





I 


2 


3 


4 


9 


II 


14 


18 


21 


25 


3o 


34 


& 


44 


5o 


56 


62 


3 


17 





I 


2 


3 


4 


6 


8 


II 


14 


17 


20 


24 


28 


33 


43 


48 


54 


60 


18 





I 




3 


4 


6 


8 


IO 


i3 


16 


20 


23 


27 


32 


36 


41 


46 


52 


58 


64 


19 





I 




3 


4 


6 


8 


IO 


i3 


16 


[I 


22 


26 


3o 


35 


40 


45 


5o 


56 


61 


20 





I 




2 


4 


5 


7 


IO 


12 


i5 


22 


25 


29 


34 


38 


43 


48 


54 


60 


21 





I 




2 


4 


5 


7 


9 


12 


i5 


18 


21 


25 


2 


33 


37 


42 


% 


52 


58 


22 


o 


I 




2 


4 


5 


7 


9 


12 


14 


n 


21 


24 


32 


36 


4i 


5i 


56 


23 





I 




2 


3 


5 


7 


9 


II 


14 


17 


20 


23 


27 


3i 


35 


4o 


45 


5o 


55 


24 





I 




2 


3 


5 


7 


9 


II 


14 


16 


20 


23 


27 


3i 


35 


39 


44 


49 


54 


25 


o 


I 




2 


3 


5 


2 


I 


II 


i3 


16 


*9 


23 


26 


3o 


34 


£ 


43 


48 


53 


26 





I 




2 


3 


5 


II 


i3 


16 


J 9 


22 


26 


3o 


34 


42 


47 


52 


27 





I 




2 


3 


5 


6 


8 


II 


i3 


16 


;g 


22 


25 


29 


33 


37 


42 


47 


52 


28 





I 




2 


3 


5 


6 


8 


IO 


i3 


16 


22 


25 


3 


33 


37 


41 


46 


5i 


29 





I 




2 


3 


5 


6 


8 


IO 


i3 


i5 


18 


21 


25 


32 


n 


41 


46 


5i 


80 





1 




2 


3 


5 


6 


8 


10 


i3 


i5 


18 


21 


25 


28 


32 


41 


45 


5o 


31 


o 


I 




2 


3 


5 


6 


8 


IO 


12 


i5 


18 


21 


24 


28 


32 


36 


4o 


45 


5o 


32 


o 







2 


3 


4 


6 


8 


IO 


12 


i5 


18 


21 


24 


28 


32 


36 


40 


45 


5o 


33 










2 


3 


4 


6 


8 


IO 


12 


i5 


18 


21 


24 


28 


32 


36 


40 


45 


49 


34 
35 












2 


3 


4 


6 


8 


IO 


12 


i5 


18 


21 


24 


28 


32 


36 


40 


45 


49 




2 


3 


4 


6 


8 


IO 


12 


i5 


18 


21 


24 


28 


32 


36 


40 


45 


49 


36 





I 




2 


3 


4 


6 


8 


IO 


12 


i5 


18 


21 


24 


28 


32 


36 


40 


45 


49 


37 


o 


I 




2 


3 


4 


6 


8 


IO 


12 


i5 


18 


21 


24 


28 


32 


36 


40 


45 


49 


88 





I 




2 


3 


4 


6 


8 


IO 


12 


i5 


18 


21 


24 


28 


32 


36 


40 


45 


5o 


39 





I 




2 


3 


4 


6 


8 


IO 


12 


i5 


18 


21 


24 


28 


32 


36 


40 


45 


5o 


40 





I 




2 


3 


5 


6 


8 


IO 


13 


i5 


18 


21 


25 


28 


32 


36 


41 


45 


5o 


41 





I 




2 


3 


5 


6 


8 


IO 


i3 


i5 


18 


21 


25 


28 


32 


37 


4i 


46 


5i 


42 


o 


t 




2 


3 


5 


6 


8 


IO 


i3 


i5 


18 


22 


25 


29 


33 


37 


41 


46 


5i 


43 





I 




2 


3 


5 


6 


8 


IO 


i3 


16 


18 


22 


25 


29 


33 


n 


42 


46 


5i 


44 





I 




2 


3 


5 


6 


8 


IO 


i3 


16 


19 


22 


25 


29 


33 


42 


47 


52 


45 


o 


I 




2 


3 


5 


■6 


8 


II 


i3 


16 


l 9 


22 


26 


3o 


34 


38 


43 


48 


53 


46 





I 




2 


3 


5 


6 


8 


II 


i3 


16 


l 9 


22 


26 


3o 


34 


38 


43 


48 


53 


47 





I 




2 


3 


5 


7 


9 


II 


i3 


16 


19 


23 


26 


3o 


35 


3 9 


44 


49 


54 


48 





I 




2 


3 


5 


7 


9 


II 


14 


n 


20 


23 


27 


3i 


35 


40 


44 


5o 


55 


49 





I 




2 


3 


5 


7 


9 


II 


i4 


17 


20 


23 


27 


3i 


36 


40 


45 


5o 


56 


50 





I 




2 


4 


5 


7 


9 


II 


14 


17 


20 


24 


28 


32 


36 


4i 


46 


5i 


n 


51 


o 


I 




2 


4 


5 


7 


9 


12 


14 


\l 


21 


24 


28 


32 


37 


42 


% 


52 


52 





I 




2 


4 


5 


7 


9 


12 


i5 


21 


25 


29 


33 


38 


43 


53 


5 9 


53 





I 




2 


4 


5 


7 


IO 


12 


i5 


18 


21 


25 


29 


34 


38 


43 


49 


54 


60 


54 





I 




2 


4 


5 


7 


IO 


12 


i5 


18 


22 


26 


3o 


34 


3 9 


44 


5o 


56 


62 

oT 


55 





I 




2 


4 


6 


8 


IO 


13 


16 


*9 


22 


26 


3i 


35 


40 


45 


5i 


U 


56 





I 




3 


4 


6 


8 


IO 


i3 


16 


19 


23 


27 


3i 


36 


4i 


46 


52 


65 


57 





I 




3 


4 


6 


8 


IO 


i3 


16 


20 


24 


28 


32 


U 


42 


48 


54 


60 


66 


58 





I 


2 


3 


4 


6 


8 


ii 


14 


17 


20 


24 


28 


33 


43 


49 


55 


61 


68 


59 





I 


2 


3 


4 


6 


8 


ii 


U 


17 


21 


25 


29 


34 


3 9 


45 


5o 


57 


63 


70 


60 





I 


2 


3 


4 


6 


9 


u 


U 


18 


22 


26 


3o 


35 


40 


46 


52 


58 


65 


72 


61 





I 


2 


3 


5 


7 


9 


12 


i5 


18 


22 


26 


3i 


36 


42 


47 


53 


60 


67 


75 


62 





I 


2 


3 


5 


7 


9 


12 


i5 


*9 


23 


11 


32 


37 


43 


49 


55 


62 


70 


11 


63 





I 


2 


3 


5 


7 


IO 


12 


16 


20 


24 


33 


39 


44 


5i 


5 7 


64 


72 


64 





I 


2 


3 


5 


7 


IO 


i3 


16 


20 


24 


29 


34 


4o 


46 


5-2 


5 9 


67 


75 


83 


65 





I 


2 


3 




I 






\l 


21 


25 


3o 


36 


4i 


48 


54 


62 


69 


1? 


86 


66 





I 


2 


3 


5 


ii 


14 


22 


26 


32 


37 


43 


5o 


57 


64 


72 


90 


67 





I 


2 


4 


6 


8 


ii 


14 


18 


23 


28 


33 


3 9 


45 


52 


5 9 


67 


76 


85 


94 


68 





I 


2 


4 


6 


8 


12 


i5 


19 


24 


2 9 


34 


4o 


47 


54 


62 


70 


11 


3 


99 


69 





1 


2 


4 


6 


9 


12 


16 


20 


25 


3o 


36 


42 


49 


5 7 


65 


74 


104 


70 





I 


2 


4 


6 


9 


13 


16 


21 


26 


32 


38 


44 


52 


60 


68 


£ 


88 


98 


HO 


71 





I 


2 


4 


7 


IO 


i3 


17 


22 


27 


33 


40 


47 


55 


63 


72 


93 


104 


Il6 


72 





I 


3 


5 


7 


IO 


U 


18 


23 


2 9 


35 


42 


49 


58 


67 


76 


87 


98 


in 


124 
... 



TABLE V. 



CONTAINING 



PARTS OF A MERIDIAN; 



OB, 



THE INCREASED LATITUDES, FOE FINDING THE MERIDIONAL DIFFERENCE 

OF LATITUDE, WHEN A SHIP'S DEPARTURE IS INCREASED 

TO HER DIFFERENCE OF LONGITUDE. 



74 



MERIDIONAL PARTS. 



LATITUDE. 


MlN. 




0° 


1° 


2° 


3 8 


4° 


5° 


6° 


7° 


8° 


9° 


10° 


11° 


12° 


MlK. 




0-0 


6o> 


I20«0 


180. 1 


240.2 


3oo-4 


36o-7 


421. 1 


481.6 


542-2 


6o3-i 


664-1 


725.3 


1 


1-0 


61 -0 


21-0 


81. 1 


41-2 


01.4 


61.7 


22. 1 


82.6 


43-3 


o4- 1 


65-i 


26.3 


1 


2 


2«0 


62-0 


22-0 


82.1 


42-2 


02«4 


62.7 


23-1 


83-6 44-3 


o5«i 


66.1 


27.4 


2 


3 


3-o 


63-o 


23-0 


83-1 


43-2 


o3-4 


63-7 


24.1 


84-6 


45-3 


06- 1 


67.1 
68.2 


28-4 


3 


4 


4-o 


64-0 


24-0 


84-1 


44-2 


04-4 


64-7 


25-1 


85-6 


46-3 


07-1 
08- 2 


29-4 


4 


5 


5.o 


65-o 


25-0 


85-1 


45-2 


o5-4 


65-7 


26-1 


86-6 


47-3 
48-3 


69.2 


3o-5 


5 


6 


6-o 


66-0 


26-0 


86-1 


46-2 


06-4 


66-7 


27.1 


87-6 


09-2 


70-2 


3i-5 


6 


7 


U 


67-0 
68-o 


27.O 
28-0 


87.1 


47-2 


07-4 

08.4 


67-7 


28.1 


88-6 


49-3 


10-2 


71.2 


32-5 


7 


8 


88-1 


48-2 


68.7 


29.1 


89-6 


5o-3 


II-2 


72-2 


33-5 


8 


9 


9.0 


69.0 


29-0 


89.1 


49.2 


09.4 


69.7 


3o-i 


90.7 


5i«4 


12-2 


73.3 


34.5 


9 


10 


I0«0 


70.0 


i3o-o 


190. 1 


250'2 


3io-4 


370-7 


43i-i 


491.7 


552-4 


6l3-2 


674.3 


735.6 


10 


11 


II«0 


71.0 


3i-o 


91. 1 


5l-2 


11. 4 


71.7 


32-1 


92.7 


53-4 


14-2 


75.3 


36-6 


11 


12 


12-0 


72-0 


32-0 


92-1 


52-2 


12.4 


72-7 


33-1 


93.7 


54-4 


i5-3 


76.3 


37-6 
38-6 


12 


13 


i3-o 


73.0 


33-0 


93.1 


53.2 


13-4 


73.7 


34-2 


94-7 


55-4 


16.3 


77.3 
78-4 


18 


14 


U-o 


74-o 


34.0 


94-1 


54-2 


14-4 


74-7 


35-2 


9 5-7 


56-4 


17.3 

18.3 


3 9 -6 


14 


15 


i5'0 


"p-O 


35-0 


9 5.i 


55-2 


i5-4 


&3 


36-2 


96-7 


57-4 
58-4 


79'4 
80.4 


40-7 


15 


16 


i6«o 


76-0 


36-0 


96.1 


56-2 


16.4 


3 7 .2 


97.7 
98.7 


19.3 


4i-7 


16 


17 


i7-o 


77.0 


37.0 
38-o 


III 


57.2 
58-2 


17-5 

i8.5 


77.8 


38-2 


59-4 


20-3 


81.4 


42-7 


17 


18 


i8-o 


78-0 


78-8 


39-2 


99.8 


6o-5 


21-3 


82.4 


43-7 
44-8 


18 


19 


19.0 


79.0 


3g« 


99.1 


59-2 


19.5 


79.8 


4o« 2 


5oo-8 


6i-5 


22-4 


83-5 


19 


20 


20«0 


8o- 


i4o«o 


200-1 


260-2 


32o«5 


38o-8 


441-2 


5oi-8 


562-5 


623.4 


684.5 


745-8 


20 


21 


2I«0 


8i-o 


4l '0 


01 • I 


6l3 


21-5 


8i-8 


42-2 


02-8 


63-5 


24-4 


85-5 


46-8 


21 


22 


22-0 


82-0 


42-0 


02« I 


62.3 


22-5 


82.8 


43-2 


o3-8 


64-5 


25-4 


86-5 


47-8 


22 


23 


23-0 


83-o 


43-o 


03- 1 


63-3 


23-5 


83-8 


44-2 


04-8 


65-5 


26.4 


87.5 


48-9 


23 


24 


24-0 


84-0 


44-o 


04-1 


64-3 


24-5 


84-8 


45-2 


o5-8 


66-6 


27-4 


88-6 


49.9 


24 


25 


25-0 


85-0 


45-0 


o5-i 


65-3 


25-5 


85-8 


46-3 


o6- 8 


67-6 


28-5 


89-6 


5o-9 


25 


26 


26.0 


86-o 


46-o 


06.1 


66-3 


26.5 


86-8 


47-3 


07-8 


68-6 


29-5 


90-6 


5i-9 


26 


27 


27>0 


87.0 


47.0 


3:; 


67.3 


27.5 


87-8 


48-3 


08-9 


69-6 


3o-5 


91-6 


53-o 


27 


28 


28-0 


88-o 


48-0 


68-3 


28-5 


88-8 


4 9 -3 


09.9 


70-6 


3i-5 


92-6 


54-o 


28 


29 


29.O 


89.0 


49.0 


09.1 


69.3 


29.5 


89-8 


5o-3 


10-9 


71-6 


32-5 


9 3- 6 


55-o 


29 


30 


3o«o 


90-0 


i5o«o 


2I0-I 


270-3 


33o-5 


390-8 


45i-3 


5n-9 


572-6 


633-5 


694-7 


756-0 


30 


31 


3i-o 


91-0 


5i«o 


II • I 


71.3 


3i-5 


91-8 


52-3 


12.9 


73-7 


34-6 


95.7 


5 7 -i 


31 


82 


32-0 


92.0 


52« 


I2-I 


72.3 


32-5 


92-9 


53-3 


13.9 


74-7 


35-6 


96-7 


58-i 


32 


83 


33-0 


93.0 


53-1 


i3-i 


73.3 


33-5 


93-9 


54-3 


14-9 


7 5-7 


36-6 


97-7 


5 9 -i 


33 


84 


34-o 


94.0 


54-i 


14-1 


74.3 


34-5 


94-9 


55-3 


i5-9 


76-7 


37-6 


98-7 


6o-i 


84 


35 


35-0 


95.0 


55-1 


i5-i 


75.3 


35-5 


95.9 


56-3 


16-9 


77'7 


38-6 


99.8 


61. 1 


85 


86 


36-o 


96-0 


56-i 


16. 1 


76.3 


36-5 


96-9 


5 7 -3 


18-0 


78-7 


3 9 - 6 


700-8 


62«2 


36 


37 


37-0 


97-0 
98-0 


5-7.1 
58-i 


17. 1 


77.3 


37-5 


97-9 


58-4 


19-0 


79.7 


40-7 


oi-8 


63-2 


87 


38 


38-o 


18. 1 


78.3 


38-5 


98-9 


5g-4 


20-0 


8o-8 


4i-7 


02-8 


64-2 


38 


39 
40 


39.0 


99.0 


59-1 


19. 1 


79.3 


3 9 -6 


99.9 


60-4 


21-0 


8i-8 


42-7 


o3-8 


65-2 


39 
40 


40. 


100 -0 


i6o- 1 


220-2 


280-3 


34o-6 


400-9 


461-4 


522-0 


582.8 


643-7 


704-9 


766.3 


41 


4i-o 


0I«0 


6i- 1 


21. 2 


8i-3 


4i-6 


01-9 


62.4 


23-0 


83-8 


44-7 


o5-9 


67.3 
68-3 


41 


42 


42-0 


02« 


62-1 


22-2 


82-3 


42-6 


02-9 


63-4 


24-0 


84-8 


45-8 


06-9 


42 


43 


43-o 


o3«o 


63-i 


23-2 


83-3 


43-6 


o3-9 


64-4 


25-0 


85-8 


46-8 


07-9 


69.3 


43 


44 


44.0 


o4-o 


64-1 


24«2 


84-3 


44-6 


04-9 


65-4 


26-0 


86-8 


47-8 


09-0 


70-4 


44 


45 


45-o 


o5-o 


65-i 


25-2 


85-3 


45-6 


o5-9 


66-4 


27-1 


87.9 


48-8 


10-0 


7i-4 


45 


46 


46-0 


06-0 


66-i 


26-2 


86-3 


46-6 


07-0 


67-4 


28-1 


88-9 


49-8 


II-O 


72-4 


46 


47 


47-o 
48-0 


07.0 


67.1 
68-i 


27-2 


87.3 


47-6 


08-0 


68-4 


29.1 


89.9 


5o-8 


12-0 


73-4 


47 


48 


08. 


28-2 


88-3 


48-6 


09-0 


69.5 


3o-i 


90-9 


5i-9 


i3-i 


74-5 


48 


49 


49.0 


09-0 


69.1 


29.2 


89.3 


49-6 


10-0 


70-5 


3i-i 


91.9 


52-9 


i4-i 


75.5 


49 


50 


5o-o 


IIO'O 


170-1 


230-2 


290-3 


35o-6 


ill-O 


471-5 


532-1 


592-9 


353-9 


7i5«i 


776-5 


50 


51 


5i-o 


II« 


71. 1 


3l-2 


9i-3 


5i-6 


12-0 


72-5 


33-i 


93.9 


54-9 


16. 1 


77.5 


51 


52 


52-0 


12-0 


72-1 


32-2 


92-4 


52-6 


i3-o 


7 3-5 


34-i 


95-o 


55-9 


17. 1 


78-6 


52 


53 


53-o 


i3-o 


7 3-i 


33-2 


9 3-4 


53-6 


14-0 


74-5 


35-i 


96-0 


57-0 


18-2 


79*6 

8o- 6 


53 


54 


54-o 


i4-o 


74-i 


34-2 


94-4 


54-6 


i5-o 


7 5-5 


36-2 


97-0 


58-o 


19-2 


54 


55 


55-0 


i5«o 


75.1 


35-2 


95-4 


55-6 


16. 


76-5 


3 7 .2 


98-0 


59-0 


20-2 


81.7 


55 


56 


56-o 


16-0 


76.1 


36-2 


96-4 


56-6 


17-0 


77-5 


38-2 


99.0 


60 -0 


21-2 


82-7 


56 


57 


57.0 


17-0 


ft! 


3 7 .2 


97-4 


5 7 -6 


18-0 


78-5 


39.2 


600 -0 


6i-o 


22-3 


83-7 


57 


58 


58-o 


18. 


38-2 


98-4 


58-6 


19.0 


79-5 


40-2 


01-0 


62-1 


23-3 


84-7 


58 


59 


5g-o 


19-0 


79.1 


39-2 


99.4 


5 9 -7 


20. 


8o.5 


41-2 


02-1 


63-1 


24-3 


85-8 


59 



MERIDIONAL PARTS. 



75 



LATITUDE. 


MlN. 




13° 


14° 


15° 


16° 


17° 


18° 


I9 a 


20° 


21° 


22° 


23° 


24° 


786.8 


848.5 


910.5 


9 ]U 


io35.3 


1098.2 


ii6i.5 


I225.I 


1289.2 


1353.7 


1418.6 


1484.1 


1 


87.8 


49-5 


iilS 


36.3 


99.3 


62.5 


26.2 


90.3 


54.8 


19.7 


85.2 


2 


88.8 


5o.5 


12.6 


74.8 


37.4 
38.4 


1 ioo.3 


63.6 


27.3 


91.3 


55.8 


20.8 


86.3 


3 


89.9 


5i.6 


i3.6 


75.9 


01.4 


64.7 


28.3 


92.4 


56. 9 


21.9 


87.3 


4 


90.9 


52.6 


14.6 


76.9 


3 9 .5 


02.4 


65.7 


29.4 


9 3.5 


58.o 


23.0 


88.4 


5 


91.9 


53.6 


i5. 7 


78.0 


4o.5 


03.5 


66.8 


3o.4 


94.5 


59,0 


24.1 


89.5 


6 


92.9 


54.7 


16.7 


79.0 
80.0 


41.6 


o4.5 


67.8 
68.9 


3i.5 


9 5.6 


60.1 


25.1 


90.6 


7 


94-0 


55.7 


i8!8 


42.6 


o5.6 


32.6 


96,7 


61.2 


26.2 


9 2 : 8 


8 


95.0 


56. 7 
57.8 


81. 1 


43.7 


06.6 


70.0 


33T6 


97.8 
98.8 


62.3 


27,3 


9 


96.0 


19.8 


8^.1 


44.7 


07.7 


71.0 


34.7 


63.4 


28.4 


93.9 


10 


797.0 
98.1 


858.8 


920.8 


983.2 


1045.8 


1 108.7 
09.8 


1172.1 


1235.8 


1299.9 


i364.5 


1429.5 


1495,0 


11 


5 9 .8 


21.9 


84.2 


46.8 


73-1 


36.8 


i3oi.o 


65.6 


3o.6 


96.1 


12 


99.1 


60.9 


22.9 


85.2 


47-9 


10.8 


74.2 


37.9 


02.0 


66.6 


31.7 

32.8 


97.2 


13 


800.1 


61.9 


23.9 


86.3 


48.9 


11.9 


75.2 


3g.o 


o3,i 


|| 


98.3 


14 


01.2 


62.9 


25.0 


87.3 
88.4 


49.9 


12.9 


76.3 


4o.o 


04.2 


33.9 


99.4 


15 


02.2 


64-0 


26.0 


5i.o 


14.0 


77-4 


4i.i 


o5.3 


69.9 


34.9 


i5oo.5 


16 


03.2 


65.o 


27.0 

28.1 


89.4 


52.0 


i5.o 


78.4 


42.2 


o6.3 


70.9 


36.o 


01.6 


17 


04.2 


66.0 


90.4 


53.i 


16.1 


79.5 


43.2 


08.5 


72.0 


37.i 
38.2 


02.7 
o3.8 


18 


o5.3 


67.1 
68.1 


29.1 


91.5 


54.i 


$ 


8o.5 


44.3 


7 3.i 


19 
20 


o6.3 


3o.i 


92.5 


55.2 


81.6 


45.4 


09.6 


74-2 


3 9 .3 


04.9 


807.3 


869.1 


931.2 


993.6 


io56.2 


1119.2 


1182.7 


1246.4 


i3io.6 


1375.3 


1440.4 


i5o6.o 


21 


08.4 


70.1 


32.2 


946 


57.3 


20.3 


83. 7 


47-5 
48.6 


11,7 


76.4 


41.5 


0-7.1 
08,2 


22 


09.4 


71.2 


33.3 


9 5.6 


58.3 


21.3 


84.8 


12.8 


77-4 


42.6 


23 


10.4 


72.2 


34.3 


9 6 -7 


59.4 


22.4 


85.8 


49-6 


13.8 


78.5 


43.7 


09.3 


24 


11.4 


7 3.2 


35.3 


97-7 


60.4 


23.4 


86.9 


5o-7 
5i.8 


14.9 


79.6 
80.7 
81.8 


44-8 


10.4 


25 


12.5 


74.3 


36.3 


98.8 


61.4 


24.5 


88.0 


16.0 


45.8 


ii.5 


26 


i3.5 


75.3 


374 


99.8 


62.5 


25.5 


89.0 


52.8 


17.1 


46.9 


12.6 


27 


14.5 


76.3 


38.4 


1000.8 


63.5 


26.6 


90.1 


53.9 


18.1 


82.8 


48.0 


i3. 7 


28 


i5.5 


77-4 


3 9 .5 


01.9 


64.6 


27.6 


91. 1 


55.o 


19.2 


83. 9 


49.1 


14.8 


29 


16.6 


78.4 


40.5 


02.9 


65.6 


28.7 


92.2 


56.o 


20.3 


85.o 


5o.2 


i5. 9 


30 


817.6 


879-4 


941.6 


1004.0 


1066.7 


1 1 29.7 


1193.2 


1257. 1 
58.2 


1321.4 


i386.i 


1451.3 


1 517.0 
18.1 


31 


18.6 


8o.5 


42.6 


o5.o 


67.7 
68.8 


3o.8 


94.3 


22.5 


8 F 
88.3 


52.4 


32 


19.6 


8i.5 


43.6 


06.1 


3:i.8 


95.4 


59.2 


23.5 


53.5 


IQ.2 


33 


20.7 


82.5 


44-7 


07.1 


69.8 


32.9 


96.4 


6o.3 


24.6 


89.4 


54.6 26.3 


84 


21.7 


83.6 


45.7 


08.1 


70.9 


34.0 


97.5 


61.4 


25.7 


90.4 


55.6 


21.4 


85 


22.7 


84.6 


46.7 


09.2 


72:0 


35.i 


98.5 


62.4 


26.7 


91.5 


56.7 

5 7 .8 


22.5 


36 


23.8 


85.6 


47-8 


10.2 


73.0 


36.1 


99.6 


63.5 


28% 


92.6 


23.6 


87 


24.8 


86.7 


48.8 


11. 3 


74-1 


3 7 .2 


1200.7 


64.6 


9 3.T 

94.8 


58. 9 


24.7 

25.8 


38 


25.8 


87.7 


49-9 


12.3 


7 5.i 


38.2 


01.7 
02.8 


65.6 


3o.o 


60.0 


39 
40 


26.9 


88.7 


59.9 


i3.4 


76.2 


3 9 .3 


66.7 


3i.i 


9 5.8 


61. 1 


26,9 


827.9 


889.8 


95i.9 


10144 


1077.2 
78.3 


1 i4o.3 


1203.9 


1267.8 


i332,i 


1396.9 


1462.2 


i528.o 


41 


28.9 


90.8 


53.o 


1 5.4 


4i.4 


04.9 


68.8 


33.2 


98.0 


63.3 


29.1 


42 


29.9 


91.8 


54.o 


i6.5 


79.3 


42.4 


06.0 


69.9 


34.3 


99.1 


64.4 


30.2 


43 


3i.o 


92.9 


55.i 


17.5 


80.4 


43.5 


07.1 


71.0 


35.3 


1400.2 


65.5 


3i.3 


44 


32.0 


93.9 


56.i 


18.6 


81.4 


44.6 


08.1 


72.1 


36.4 


oi,3 


66.6 


32.4 


45 


33.o 


94.9 


fr 1 


19.6 


82.5 


45.6 


09.2 


7 3.i 


37.5 


02.4 


M 


33.5 


46 


34.1 


96.0 


58.2 


20.6 


83.5 


46.7 


10.2 


74.2 


38.6 


o3.4 


34.6 


47 


35.i 


9,7.0 


59.2 


21.7 


84.6 


47-7 
48.8 


II.3 


75.3 


39.7 


04.5 


69.8 


35.7 
36.8 


48 


36.i 


98.0 


60.2 


22.7 
23.8 


85.6 


12.4 


76.3 


46.7 


o5.6 


70.9 


49 


3 7 .2 


99.1 


6i.3 


86.7 


49.9 


i3.4 


77-4 


41.8 


06,7 


72.0 


37.9 


50 


838.2 


900.1 


962.3 


1024.8 


1087,7 
88.8 


1 1 50.9 


1214.5 


1278.5 


1342.9 


1407.8 
08,8 


1473.1 


1539.0 


51 


39.2 


01. 1 


63.4 


25.9 


52.0 


i5.5 


79.5 
80.6 


44.o 


74.2 


40.1 


52 


40.2 


02.2 


64.4 


26.9 


89.8 


53.o 


16,6 


45.1 


09.9 


75.3 


41.2 


53 


41.3 


03.2 


65.5 


28.0 


90.9 


54.1 


n-7 


81.7 


46.1 


11, 


76.4 


42.3 


54 


42.3 


04.3 


66.5 


29.0 


91.9 


55.1 


18.7 


82.8 


47-2 


12. 1 


78.6 


43.4 


55 


43.3 


o5.3 


67.5 
68.6 


3o.i 


93.0 


56.2 


19.8 


83.8 


48.3 


13.2 


44. J) 


56 


444 


o6.3 


3i.i 


940 


57.2 


20.9 


84.9 


49.4 


14.3 


79,7 
80.8 


45.6 


57 


45.4 


074 


69.6 


32.2 


95.1 


58.3 


21.9 


86.0 


5o.4 


1 5.4 


46.7 


58 


46.4 


08.4 


70.7 


33.2 


96.1 


5 9 .4 


23.0 


87.0 
88.1 


5i.5 


16.5 


81.9 


47-8 
48.9 


59 


47.5 


09.4 


71.7 


34.3 


97.2 


60.4 


24.1 


52.6 


17.5 


83.o 



13* 



76 



MERIDIONAL PARTS. 



LATITUDE 



Mm. 


1 

2 



10 
11 
12 
13 
14 
15 
16 
17 
18 
19 

20 
21 
22 
23 
24 
25 
26 
27 
28 
29 

SO 
31 
32 
33 
34 
35 
36 
37 
88 
39 

40 
41 
42 
43 
44 
45 
46 
47 
48 
49 
50 
51 
52 
53 
54 
55 
56 
57 
58 
59 



25° 26° 27 



i55o' 
5i. 

5 2 . 
53. 
54' 
55. 
56- 

5 7 - 
58. 
5 9 - 

i56i- 

62- 

63- 
64- 
65. 
66- 

69. 
71. 



1616. 

\l 

19. 

20. 
22- 
23- 

24- 

25. 
26. 

1627- 
28" 

3i. 

32- 

33. 
34' 
35. 
36. 
37' 



1572. 
7 3. 

t 

]l 

II 
82 

i583 
84 

85 



87-6 



i683- 
84- 

85- 
86. 



90 
91 

9 3 

1694 
9 5 

U 

99 

1700 

01 

02 
o3 
04 



i638. 
3 9 - 
4i' 

42' 

43. 

44' 
45. 
46. 



1649 
5i 

52 

53 
54 
55 
56 

ll 

60 



1751- 

52- 

53- 
54- 
55. 
56- 
58- 

?" 
60 • 

61. 

1762- 
63. 

64- 
65- 

k 

69. 
70. 
71. 
72. 



706- 
07. 

08- 
09. 

10- 

12. 
i3- 
i5< 

16. 

1717- 
18. 
19. 
20. 

21. 

22' 
24' 
25. 
26> 

27 



28 c 



1773. 
75. 
76. 

$ 

81. 

83- 
84. 

1785. 
86- 

II 
89. 

90. 

9 J' 

9 4 

9 5 



29 c 



1819 
20 
21 

22 
24 

25 

26 

3 

29 
i83o 

32 

33 
34 
35 
36 

U 

4o 
41 



1842 
43. 
44. 
45 
46. 
48 

i 9 

5o 
5 1 

52 

1 853 
55 
56 

u 
s 

61 

63 
64 



30 c 



90 
91 
9 3 

9 i 
f, 

11 

1899. 
1901. 

02- 

03. 

o4' 
o5. 
06. 
08' 
09. 

10' 



1911 • 

12- 

i3- 
i5- 
16- 

n- 
18. 
19. 

20- 
21- 

1923- 

24' 
25. 

26. 

II- 

3o 
3i 

32 

33 



zr 



1958. 

60 • 
61. 
62. 

63. 
65. 
66- 

3: 

1969. 
70. 

73. 
%' 

76. 

77' 

& 



2028- 

3o- 

3i. 
33. 
34- 
35. 
36. 

3 9 . 

2040. 
4i" 

42' 

43. 
44- 

46. 

t 
& 



90. 
92. 

1993. 
94' 

9 l' 
96. 

97' 

99 

2000 

01 

02 
o3 



32° 



33° 



2099 
2100 
01 
o3 
04 
o5 
06 
07 
09 
10 

2111 
12 

i3 
i5 

16 

17 
18 

19 

21 

22 



2052- 

53- 

54- 
55. 
56- 
58- 

60 ■ 
6i- 
62. 

2o63. 
65. 

66^ 

!?: 

6 9 
71 

73 
74 



2171-5 

72.7 
7 3-9 
75.1 
76-3 
77.5 
78.7 
8o-o 
81-2 
82.4 

2183-6 
84-8 
86-o 
87-2 
88-4 
89-6 
90-8 
92-0 
9 3-3 
94.4 



2123- 

24- 

25- 

27- 

28- 

29. 

3o- 
3i- 
33- 
34- 

2i35- 
36- 

3 9 - 
40. 
4i- 

42' 

43. 
45. 
46. 



34 c 



35° 

2244-3 
45-5 
46-8 
48-0 
4 9 -2 
5o-4 
5i-6 
52- 9 
54-i 
55-3 



2256-5 
5 7 - 8 
5 9 -o 
60-2 
61-4 
62-7 
63- 9 
65-i 
66-3 
67-5 



2I 9 5- 

96. 
98. 

99. 
2200- 

01- 

o3- 
04- 
o5- 
06. 

2207. 
09. 
10 
11 
12 
i3 
i5 
16 

i 



2268-8 
70-0 
71.2 
72-5 
7 3-7 
74-9 
76-1 
77-4 
78-6 
79.8 

2281-0 
82-3 
83-5 

84-7 
86-o 
87-2 
88-4 
89.7 
90-9 
92-1 



i5 9 4- 

& 

% 

99. 
1600- 

02- 

o3. 
04. 

i6o5. 
06. 

09 
10 
12 
i3 

14 
i5 



1661- 
62- 
63- 
64- 
65- 
66- 

69. 
70- 
71. 

1672. 
7 3. 

7 i 

76 

77 

1 9 
80 

81 

82 



1728- 

29' 
3o- 
3i- 
33- 
34- 
35- 
36- 

II 

i 7 3 9 - 
4i- 
42- 
43- 
44. 
45- 
46- 
47- 



5o 



1796- 

s: 

1800- 

01- 
02- 

o3- 
04- 
o5- 

06. 

1808. 
09. 

IO- 
II- 
12- 

i3- 

14. 

16. 



1 865- 
66- 

H: 

69. 

71. 

72- 

7 3- 

7 5. 

i-8 7 6- 
7 8- 
79. 
80. 
81. 
82. 

83. 



1934- 
35- 

II 

39- 
40- 
41- 
42- 
44- 
45- 

1946- 
47. 
48- 

P, 

52- 

53. 

54. 

i 55 

56. 



2004-9 
06-0 

08-4 

09-6 

10 

11.9 

i3 

14 

i5-4 

2016-6 
17-8 
19-0 

20-2 
21-3 
22-5 

23.7 
24-9 
26-0 
27-2 



2075. 
76. 
78. 

79 
80 
81 
82 



7 

9 

1 

3 

5 

7 

9 

84-o 

85-2 

86-4 

2087-6 



90 

91-2 

92-4 

9 3- 6 

94-8 

96-0 

97-i 



214 
4 
49.8 
5i-o 

52 

53 

54-6 

55-8 

I 

2159-4 
60 -7 
61-9 
63 

64 
65 

66 
67 
69 
70 



2219 
21 

22 
23 
24 
26 

II 
It 

2232 

33 
34' 
35. 

37 
38 

39 
40 
41 
43 



2293.3 
9 4-6 
9 5- 8 
97.0 
98-3 
99.5 

23oo-7 

02-0 
03-2 

04.4 

23o5-7 
06-9 
08-1 
09.4 
io-6 
n-8 
i3-i 
14-3 
i5-5 
16-8 



MEEIDIONAI, PARTS. 



77 



LATITUDE. 


MlN. 




36° 


37° 


38° 


39° 


40° 


41° 


42° 


43° 


44° 


45° 


46° 


MlN. 


23l8.0 2 


392.62468.32545.0 


'622.7 


2701.6 


2781.7: 


863. 1 : 


945.8 3o3o.o3n5.6 





1 


19.2 


93.9 


69.5 


46.2 


24-0 


02.9 


83.i 


64.5 


47.2 


3i.4 


18!? 


1 


2 


20.5 


95.1 


70.8 


47-5 


25.3 


04-3 


84.4 


65.8 


48.6 


32.8 


2 


3 


21-7 


96.4 


72.1 


48.8 


26.6 


o5.6 


85.8 


67.2 


5o.o 


34.2 


19.9 


3 


4 


23.o 


97-7 
98.9 


73.4 


5o.i 


27.9 


06.9 
o8.3 


87.1 


68.5 


5i.4 


35.6 


21.4 


4 


5 


24-2 


74.6 


5i.4 


29.2 


88.5 


70.0 


52.8 


37.0 


22.8 


5 


6 


25.4' 


2400.2 


75.9 


52.7 


3o.5 


09.6 


89.8 


7 i.3 


54.2 


38.4 


24.2 


6 


7 


26.7 


01.4 


77.1 


54-0 


31.9 


10.9 


91.2 


72.7 


55.6 


3 9 .8 


26.7 


7 


8 


27.9 


02.7 


78.5 


55.3 


33.2 


12.2 


92.5 


74-1 


57.0 
58.3 


4i.3 


27.1 


8 


9 


29.1 


03.9 


79-7 


56.6 


34.5 


i3.5 


9 3.8 


75.4 


42.7 


28.5 


9 


10 


2330.4 


24o5.2 


2481.0 


2557.8 


2635.8 


27U.9 


2795.1 


2876.8 


2959.8 


3o44. 1 


3i3o.o 


10 


11 


3i.6 


06.4 


82.2 


59.1 


37.i 
38.4 


16.2 


96.5 


78.2 


61. 1 


45.5 


3i.5 


11 


12 


32.9 


07.7 


83.5 


694 


18.9 


97-9 
99.3 


29-5 


62.5 


47.0 
48.4 


32.9 
34.3 


12 


13 


34.i 


09.0 


84.8 


61.7 


39.7 


80.9 


63. 9 
65.3 


13 


14 


35.3 


10.2 


86.1 


63.0 


41.0 


20.2 


2800.6 


82.3 


49.8 


35.8 


14 


15 


36.6 


n.5 


87.4 
88.6 


64.3 


42.3 


21.5 


02.0 


83.7 


66.7 


5l.2 


37.2 

38.7 


15 


16 


3 7 .8 


12.7 


65.6 


43.6 


22.9 


o3.3 


85.0 


68.1 


52.6 


16 


17 


39-°, 


14.0 


89.9 


66.9 


44-9 
46.3 


24.2 


04.7 


86.4 


69.5 


54.1 


40.1 


17 


18 


4o.3 


l5.2 


91.2 


' 68.2 


25.5 


06.0 


87.8 


70.0 
72J 


55.5 


41.6 


18 


19 


4i.5 


i6.5 


92.4 


69.5 


47-6 


26.8 


07.3 


89.1 


56.9 


43.o 


19 


20 


2342.8 


2417.8 


2493.7 


2570.7 


2648.9 


2728.2 


2808.8 


2890.5 


2973.7 


3o58.3 


3U4.5 


20 


21 


44.0 


19.0 


95.0 


72.0 


5o.2 


29.5 


10. 1 


91.9 


7 5.i 


59.7 


45.9 


21 


22 


45.3 


20.3 


96.3 


73.3 


5i.5 


3o.8 


11.4 


9 3.3 


76.5 


61.2 


47-4 
48.8 


22 


23 


46.5 


21.5 


97.6 


74.6 


52.8 


32.2 


12.8 


94.7 


77-9 
79.3 


62.6 


23 


24 


47-7 


22.8 


98.8 


75.9 


54.1 


33.5 


14.1 


96.0 


64.0 


5o.3 


24 


25 


49.0 


24.0 


25oo.i 


77.2 


55.5 


34.8 


i5.5 


97-4 


80.7 


65.4 


5i.7 


25 


26 


5o.2 


25.3 


01.4 


78.5 


56.8 


36.2 


16.8 


98.8 


82.1 


66.9 


53.2 


26 


27 


5i.5 


26.5 


02.7 


r, 


58.i 


37.5 
38.8 


18.2 


2900.2 


83,5 


68.3 


54.6 


27 


28 


52.7 


27.8 


03.9 


5 9 .4 


19.5 


oi.5 


84.9 


69.7 


56.i 


28 


29 


54.0 


29.1 


05.2 


82.4 


60.7 


40.2 


20.9 


02.9 


86.3 


71. 1 


57.5 


29 


30 


2355.2 


243o.3 


25o6.5 


2583.7 


2662.0 


2741.5 


2822.3 


2904.3 


2987.7 


3072,6 


3i59.o 


30 


31 


56.5 


3i.6 


07.8 


85.o 


63.3 


42.9 


23.6 


05.7 


89.1 


74.0I 60.4 


31 


32 


57.7 


32.9 


09.0 


86.3 


64.6 


44-2 


25.0 


084 


90.5 


75.4 61.9 


32 


33 


58. 9 


34-i 


io.3 


87.6 


66.0 


45.5 


26.3 


91.9 


$3 


63.3 


33 


34 


60.2 


35.4 


LI .6 


889 


67.3 


46.9 


27.7 


09.7 


93.3 


64.8 


34 


85 


61.4 


36. 7 


12.9 


90.2 


68.6 


48.2 


29.0 


11. 2 


94-7 


It! 


66.2 


35 


36 


62.7 


37.9 


14.2 


91.5 


69.9 


49.5 


3o.4 


12.6 


96.1 


67.7 


36 


37 


63. 9 


39.2 


l5.4 


92.8 


71.2 


50.9 


3i. 7 


14.0 


97.5 
98.9 


82.6 


69.1 


37 


38 


65.2 


40.4 


16 7 


94.1 


72.5 


52.2 


33.1 


i5.3 


84.0 


70.6 


38 


39 
40 


66.4 


41.7 


18.0 


9 5.4 


73.9 


53.5 


34-5 


16.7 


3ooo.3 


85. 4I 72.0 


39 


236 7 .6 


2443.0 


25i9.3 


2596.7 


2675.2 


2754.9 


2835.8 


2918.1 


3ooi,8 


3o86-9'3i73.5 


40 


41 


68. g 


44.2 


20.5 


98.0 


76.5 


56.2 


3 7 .2 
38.6 


19.5 


o3.2 


88.3 


75.0 


41 


42 


70.2 


45.5 


21.8 


99.3 


77.8 


5 7 .6 
58. 9 


20 9 

22.3 


04.6 


89.7 


76.4 


42 


43 


li.4 


46.8 


23.1 


2600.5 


70.1 
8o.5 


39.9 


06.0 


91.2 


77-9 
79.3 


43 


44 


72.6 


48.0 


24.4 


01.9 


60.2 


4i.3 


23.6 


07.4 


92.6 


44 


45 


73.9 


49.3 


25." 


03.2 


81.E 


6i.5 


42.6 


25.0 


08,8 


94.0 


80.8 


45 


46 


7 5.i 


5o.6 


27.C 


04.5 


83.i 


62.9 
64.3 


44.0 


26.4 


10.2 


95,5 


82.3 


46 


17 


76.4 


5i.£ 


28.3 


o5.g 


84.4 


45.4 


27.8 


1 1 ,6 


96,9 
98J 


83.7 


47 


48 


77.6 


53.i 


2Q.£ 


07.1 
08.4 


85.- 


65.6 


467 


29.2 


i3.c 


85.2 


48 


49 


78-9 


54.^ 


3oi 


87.1 


66.9 


48.1 


3o.6 


14.4 


99-7 


86.6 


49 


50 


238o.i 


2455.6 


) 2532.1 


2609.- 


2688.; 


v 2768J 


2849.5 


2932.0 


3oi5.8 


3IOI.2 


3i88.i 


50 


51 


81.4 


56.c 


) 33.; 


t Il.C 


) 89.- 


69.6 


5o.S 


33.3 


17.5 


02.6 


89.6 


51 


52 


82.6 


> 58.] 


34-- 


I2.C 


91. c 


) 71.C 


52.2 


34-7 


18.- 


04.1 


91.C 


52 


53 


83.c 


) 5g .1 


1 36.c 


) i3.t 


> 92.; 


1 72. c 


53.5 


36.i 


20.: 


o5.6 


92.' 


53 


54 


85.1 


60.' 


3 7 .: 
) 38.^ 


i4-c 


> 9 3 -' 


73.- 


54.9 
> 56.3 


3 7 .5 


21.' 


07. c 
> 08.4 


94. c 


54 


55 


86.; 


y 6l.C 


) 16.; 


95. c 


) 75.C 


38.9 


22.C 


9 5.' 


t 55 


56 


87H 


) 63. 


1 3 9 .i 


1 n.f 


) 96 V 


i 76.: 


5 7 .- 


40.C 


24.: 


09. £ 


96.C 


) 56 


57 


88.c 


> 64. 


>1 41. 


18.1 


] 91 .( 


) 77.- 


59.C 


4i.- 


25.' 


iirf 


9 8u 


i 57 


58 


90.' 


! 65. f 


1 42.. 


i 20. 


99-< 


) 79-< 


) 60. c 


43.1 


27. 


12.' 


F 99-* 


J 58 


59 


qiu 


i 6 7 .( 


) 43. ( 


) 2i.i 


£2700.* 


) 80./ 


j. 61.- 


44-^ 


V 28.1 


) 14. 


3201.. 


t 59 

- 



78 



MERIDIONAL PABTS. 



LATITUDE. 


MlN. 




47° 


48° 


49° 


50° 


51° 


52° 


53° 


54° 


55° 


56° 


57° 


3202-- 


3291-5 


3382-1 


3474-513568-8 


3665-2 


3763.6 


3864-6 


3968- c 


4073-9 


4182-6 


1 


04-2 


93-o 


83-6 


76-0) 70-4 


66-8 


65-4 


66-3 


69-7 


75.7 


84.5 


2 


o5-7 


94-5 


35-1 


77-61 72-0 


68-4 


67., 


68- 


7i-5 


77-5 


86-3 


3 


07-I 
o8-6 


96-0 


86.7 


79-1 
80-7 


73-6 


70-1 


68-fi 


69-7 

71-5 


7 3- 2 


79-3 


88-1 


4 


97-5 


88-2 


75-2 


?{'Z 


7o-4 


75-c 


8i-i 


90-0 


5 


IOI 


99.0 


89.7 


82-3 


76-8 


73.3 


72.1 


7 3-2 


76.7 


82-9 


91-8 


6 


ii. 5 


33oo«5 


91 -3 


83-8 


73-4 


75 -0 


73-7 


74-9 


78-4 


84-7 


fa 


7 


i3-o 


02-0 


92-8 


85-4 


III 


76-6 


7 5-4 


76-6 


8o- 2 


86-4 


8 


14-5 


o3-5 


94.3 


87-0 


78.2 


77.1 


78-3 


82-0 


88-2 


97.3 


9 


i5-9 


o5-o 


9 5- 8 


88-5 


83-i 


79.8 


78-7 


8o- 


83-7 


90-0 


99.2 


10 


32i7-4!33o6-5 


33 9 7-4 


3490-1 


3584-7 


368i-5 


3780-4 


388i-7 


3 9 85-4 


4091-8 


4201-0 


11 


if.Q 


08. 


98.9 


91 -6 


86-3 


83-i 


82-1 


83-4 


87-2 


9 3-6 


02-9 


12 


20-3 


09-5 


3400-4 


93-2 


87.9 


84-7 


83-7 


85-i 


88-9 


95-4 


04-7 
06-6 


13 


21- tf 


II-O 


02 -0 


94-7 


89.5 


86-4 


85-4 


86-8 


90-7 


97-2 


14 


23-3 


12-5 


o3-5 


96-3 


91. 1 


88-o 


87- 1 


88-5 


92-5 


99-0 


08-4 


15 


24-8 


i4-o 


o5-o 


97-9 


92-7 


89-6 


88-8 


90-2 


94-2 


4ioo-8 


io-3 


16 


26-2 


i5-5 


06. 6 


99.4 


94-3 


9i-3 


90-4 


92-6 


96-0 


02-6 


12- 1 


17 


27.7 


17.0 


08. 1 


35oi-o 


95.9 
97-5 


92-0 
94-5 


92-1 


9 3-7 


97-7 


o4-4 


14-0 


18 


29.2 


i8-5 


09-6 


02-6 


9 3- 8 


95-4 


99-5 


06-2 


i5-8 


19 


3o-7 


20-0 


ill 


04-1 


99.1 


96-2 


95.5 


97.1 


4001 • 2 


08-0 


17.7 


20 


3232-1 


3321-5 


3412-7 


35o5-7 


36oo-7 


3697.8 


3797-1 


38 9 8- 8 


4oo3 • 


4109-8 


4219-5 


21 


33-6 


23-0 


14-2 


07-3 


02-3 


99-4 


98-8 


3900-5 


04-7 


n-6 


21-4 


22 


35-i 


24-5 


15-7 


08- 8 


o3-o 


3701-1 


38oo-5 


02-2 


o6-5 


i3-4 


23-2 


23 


36 6 


26-0 


17.3 


io-4 


o5-5 


02-7 


02 -2 


04 -o 


o8-3 


l5-2 


25-1 


24 


38-o 


27-5 


i8-8 


120 


0,7.1 

08-7 


04-4 


o3-8 


o5-7 


IO-O 


17-1 


27-0 


25 


3 9 -5 


29-0 


20-4 


i3-5 


06-0 


o5-5 


07-4 


H-8 


18-9 


28-8 


26 


4i-o 


3o-6 


21 -9 

23-5 


i5i 


io-3 


07-6 


07-2 


09-1 


i3-5 


20-7 


3o-6 


27 


42-5 


32-1 


16.7 


11.9 


09-3 


08-9 
io-o 


io-8 


i5-3 


22-5 


32-5 


28 


44.0 


33-6 


25-o 


i8.3 


i3-6 


10-9 


12-5 


n-i 


24-3 


34-4 


29 


45-4 


35-i 


26-5 


19-8 


i5-i 


12-6 


12-2 


14-3 


18-8 


26-1 


36-2 


30 


3246-9 


3336-6 


3428-0 


3521-4 


36i6. 7 


3714-2 


38i3-9 


3916-0 


4020-6 


4127-9 


4238-1 


31 


48-4 


38-i 


29-6 


23-0 


18-4 


i5-8 


i5-6 


17-7 


22-4 


29-7 


4o-o 


32 


49-9 


39-6 


3i-i 


24-6 


20-0 


17-5 


i 7 -3 


19-4 


24-1 


3i-5 


4i-8 


33 


5,-4 


41-1 


32.7 


26-1 


21-6 


19-1 


18-9 


21-2 


25-9 


33-3 


43-7 


34 


52-8 


42-6 


34-2 


27.7 


23-2 


20-8 


20-6 


22-9 


27.7 


35-2 


45-5 


35 


54-3 


44-1 


35-8 


29-3 


24-8 


22-4 


22-3 


24-6 


29-4 


37-0 


47-4 


36 


55-8 


45-7 


3 7 -3 


3o-8 


26-4 


24-1 


24-0 


26-3 


3l-2 


38-8 


49.3 


37 


5 7 -3 


47-2 


38-8 


32-4 


28-0 


25-7 


25-7 


28.1 


33-o 


4o-6 


5i-i 


38 


58-8 


48-7 


40.4 


34-o 


29-6 


27-4 


27-4 


29-8 


34-8 


42-4 


53-o 


39 


6o-3 


5o-2 


41-9 


35-6 


3,1-3 


29-0 


29-1 


3i-5 


36-5 


44-2 


54-9 


40 


3261-7 


335i-7 


3443-5 


3537-1 


3632-8 


3730-7 


383o-8 


3933-2 


4o38-3 


4146-1 


4256-7 


41 


63-2 


53-2 


43-0 


38-7 


34-4 


32-3 


32-4 


35-o 


4o-i 


47*9 


58-6 


42 


64-7 


54-7 


46-6 


4o-3 


36-i 


34-o 


34-i 


36-7 


4i-8 


49-7 


6o-5 


43 


66-2 


56-2 


48-1 


4i-Q 
43-5 


3 7 -7 


35-6 


35-8 


38-4 


43-6 


5i-5 


62-3 


44 


67.7 


57-8 


49-7 


3 9 -3 


37-3 


3 7 -5 


40-2 


45-4 


53-4 


64-2 


45 


69-2 


5 9 -3 


5l -2 


45-o 


40-9 38-9 


39-2 


4i-9 


47-2 


55-2 


66-1 


46 


70-7 


6o- 8 


52-8 


466 


42-5 


40-6 


40-9 


43-6 


49-0 


57-0 


68-o 


47 


72-1 


62-3 


54-3 


48-2 


44-i 


42-2 


42-6 


45-4 


5o-7 
52-5 


58-8 


69-8 


48 


7 3-6 


63-8 


55-8 


49.8 


45-8 


43.9 
45.5 


44-3 


47-i 


60-7 


7'' 7 


49 


7 5-i 


65-4 


5 7 -4 


5i-4 


47-4 


46-0 


48-8 


54-3 


62.5 


73-6 


50 


3276-6 


3366-9 


3458-9 
6o-5 


3553-0 


3649-o'3747-2 


3847-7 


395o-6 


4o56-i 


4i64-3 


4275-5 


51 


78-1 


68-4 


54-6 


5o-6 


48-8 


49.4 


52-3 


57.8 


66-1 


77-4 


52 


79.6 


69-9 


62-0 


56-i 


52-2 


5o-5 


5i-i 


54-o 


5 9 -6 


68-o 


ll\ 


53 


8i-i 


7i-4 


63-6 


57.7 


53-8 


52-1 


52-8 


55-8 


6i-4 


69-8 


54 


82-6 


73-o 


65-2 


5 9 -3 


55-5 


53-8 


54-4 57-5 


63-2 


7 i-6 


83-o 


55 


84-i 


74-5 


66-7 


60-9 
62-5 


5 7 - 1 


55-5 


56-i 


5 9 -3 


65 -o 


73.5 


84-9 


56 


85.6 


76-0 


68-3 


58-7 


5 7 - 1 

58-8 


5 7 -8 


6i-o 


66-8 


75.3 


86-8 


57 


87-1 
88-5 


77-5 


69-8 


64-0 


6o-3 


5 9 -5 


62-7 


68-5 


77-1 


88-6 


58 


80.6 


7i-4 


65-6 


61-9 


6o-4 


6l-2 


64-5 


70-3 


B:S 


90-5 


59 


, 9°'° 


73-o 


67*2 


63-6 62-1 


62-9 


66-2 


72-1 


92-4 



MERIDIONAL PARTS. 



79 



LATITUDE. 


MlN. 




58° 


59° 


60° 


61° 


62° 


63° 


64° 


65° 


66° 


67° 


68° 


4294-3 


4409-1 


4027-4 


4649-2 


4775-0 


4904-9 


5o39>4 


5i 7 8-8 


5323-5 


5474-0 


563o-8 


1 


96-2 


1 1 • 1 


29-4 


5i-3 


77.1 


07.1 


4i-7 


81 


2 


26-0 


76 


6 


33 


5 


2 


98-1 


i3-o 


3i-4 


53-4 


79-3 
8i-4 


09-4 


44-o 


83 


5 


28.4 


79 


1 


36 


2 


3 


43oo • 


i5«o 


33-4 


55-4 


11. 6 


46-3 


85 


i 


3o-9 


81 


7 


38 


8 


4 


01-9 


16-9 


35-4 


57.5 


83-5 


i3-8 


48-6 


88 


33-4 


84 


3 


4i 


5 


5 


o3-7 


18.9 

20-8 


37-4 


5 9 - 6 


85-6 


16. 


5o«9 


90 


7 


35-8 


86 


9 


44 


2 


6 


o5-6 


39.4 


6i-6 


87.8 


18.2 


53-1 


9 3 


1 


38-3 


89 


4 


46 


9 


7 


07-5 


22-8 


4i-4 


63-7 
65-8 


89.9 


20-4 


55-4 


9 D 


4 


40.8 


92 





49 


6 


8 


09-4 


24-7 


43.4 


92.1 


22-6 


5 7 - 7 


97 


8 


43-2 


94 


5 


52 


3 


9 


ii-3 


26-7 


45.4 


67.8 


94-2 


24-8 


6o-o 


5200 


2 


45.7 


97 


1 


54 


9 


10 


43i3-2 


4428.6 


4547-4 


4669 • 9 


4796.3 


4927-0 


5o62-3 


5202 


6 


5348-2 


5499 


7 


565 7 


6 


11 


i5-i 


3o-6 


49.4 


72-0 


98.5 


29-2 


64-6 


04 


9 


5o«7 


55o2 


3 


60 


3 


12 


17.0 
18-9 

20-8 


32-5 


5i-4 


74-i 


4800-6 


3i-5 


66.9 


°7 


3 


53-i 


04 


9 


63 





13 


34-5 


53-5 


76., 


02-8 


33-7 


69-2 


09 


7 


55-6 


07 


4 


65 


7 


14 


36-4 


55-5 


78-2 


04-9 


35-9 


71-5 


12 


1 


58-1 


10 





68 


4 


15 


22-7 


38-4 


5 7 .5 


8o.3 


07.1 


38-i 


73-8 


14 


5 


6o-6 


12 


6 


7i 




16 


24-6 


4o-3 


5 9 .5 


82-4 


09-2 


40-3 


76*1 


16 


I 


63-i 


i5 


2 


73 


8 


17 


26.5 


42-3 


6i.5 


84-5 


11.4 


42-6 


78.4 
80.7 


19 


65-6 


n 


8 


76 


5 


18 


28.4 


44-2 


63-5 


86-5 


i3.5 


44-8 


21 


6 


68-o 


20 


4 


79 


2 


19 
20 


3o-3 


46-2 


65-6 


88-6 


i5. 7 


47-0 


83-o 


24 





70.5 


23 



6 


81 
5684 


9 

6 


4332-2 


4448-2 


4567.6 


4690.7 


4817.8 


4949-2 


5o85-3 


5226 


4 


5373-0 


5525 


21 


34-i 


5o-i 


69.6 


92-8 


20-0 


5i-5 


87-6 


28 


8 


75.5 


28 


1 


87 


3 


22 


36-o 


52-1 


7.-6 


94-9 


22-1 


53.7 


90-0 


3i 


2 


78-0 


3o 


7 


90 





28 


37-9 
3g-8 


54-i 


73.6 


97.0 


24-3 


55-9 


92-3 


33 


6 


8o-5 


33 


3 


92 


7 


24 


56-o 


75.7 


99-1 


26.4 


58-2 


94-6 


36 





83-o 


35 


9 


95 


5 


25 


41-8 


58-o 


77-7 


4701. 1 


28-6 


6o-4 


96.9 


38 


4 


85-5 


38 


6 


98 


2 


26 


43-7 
45.6 


6o«o 


79*7 


o3-2 


3o-8 


62-6 


99-2 


40 


8 


88-0 


41 


2 


5700 


9 


27 


61-9 


81.7 


o5-3 


32.9 


64-9 


5ioi-5 


43 


2 


90-5 


43 


8 


o3 


6 


28 


47-5 


63-9 


83-8 


07-4 


35-1 


67-1 


o3-8 


45 


7 


93.0 


46 


4 


06 


3 


29 


49.4 


65-9 


85-8 


09.5 


37.3 


69-4 


06 -2 


48 


1 


9 5-5 


49 





09 


1 


SO 


435i-3 


4467-8 


4587.8 


4711-6 


4839.4 


4971-6 


5io8-5 


525o 


5 


5398-0 


555i 


6 


5711 


8 


31 


53-2 


69.8 


89.9 


i3. 7 


4i-6 


7 3-8 


io-8 


52 


9 


54oo • 5 


54 


2 


14 


5 


32 


55-1 


71.8 


91-9 


15-8 


43-8 


76.1 


i3i 


55 


3 


o3-o 


56 


8 


17 


3 


33 


i 1 ' 1 


73-7 


93.9 


17.9 


45-9 


78-3 


i5-5 


57 


7 


o5-5 


5 9 


4 


20 





34 


39-0 


75.7 


96-0 


2o« 


48-1 


8o- 6 


17.8 


60 


1 


08. 1 


62 


1 


22 


7 


35 


60 • 9 
62-8 


77-7 


98-0 


22«I 


5o-3 


82-8 


2-0>I 


62 


6 


io-6 


64 


7 


25 


5 


86 


in 


4600-0 


24.2 


52«4 


85-i 


22-4 


65 





i3-i 


67 


3 


28 


2 


37 


64-7 


02-1 


26.3 


54-6 


87.3 


24-8 


67 


4 


i5-6 


69 


9 


3o 


9 


38 


66.7 


83-6 


04-1 


28.4 


56-8 


89-6 


27-1 


69 


8 


18. 1 


72 


6 


33 


7 


39 


68-6 


85-6 


06 ■ I 


3o-5 


59.0 


91-8 


29.4 


72 


2 


20-6 


75 


2 


36 


4 


40 


4370-5 


4487-6 


4608-2 


4732-6 


4861 • 1 


4994-1 


5i3i-8 


5274 


7 


5423 • 2 


55 77 


8 


5 7 3g 


2 


41 


72-4 


89-6 


10-2 


34-7 


63-3 


96.3 


34-i 


77 


1 


25-7 


80 


4 


4i 


9 


42 


74-3 


91 «5 


12-3 


36-8 


65-5 


98-6 


36-5 


79 


5 


28-2 


83 


1 


44 


7 


43 


76.3 


9 3-5 


14-3 


39-0 


67.7 


5ooo-8 


38-8 


82 





3o-8 


85 


7 


47 


5 


44 


78-2 
8o-i 


95.5 


16-4 


4i- 1 


69-9 


o3i 


411 


84 


4 


33-3 


88 


4 


5o 


2 


45 


97.5 


18-4 


43-2 


72-0 


o5-4 


43-5 


86 


8 


35-8 


9 1 





52 


9 


46 


82-1 


99.5 


20 5 


45-3 


74-2 


07-6 


45-8 


89 


3 


38-4 


93 


6 


55 


I 


47 


84-0 


45oi-5 


22-5 


47-4 


76-4 


09.9 


48-2 


9 1 


7 


40.9 


96 


3 


58 


48 


85-9 
87-8 


o3-4 


24.6 


49.5 


78-6 
8o-8 


12-2 


5o-5 


94 


1 


43-4 


98 


9 


61 


2 


49 


o5-4 


26.6 


5i.6 


14-4 


52-9 


96 


6 


46-0 


56oi 


6 


64 





50 


4389-8 


4507-4 


4628.7 


4753-7 


4883-0 


5oi6-7 


5i55-2 


5299 





5448-5 


56o4 


2 


5 7 66 


8 


51 


91-7 


09.4 


3o-7 


55-9 


85-2 


18.9 


5 7 .6 


53oi 


5 


5i -o 


06 


9 


69 


5 


52 


9 3- 6 


11. 4 


32-8 


58-o 


87-4 


21-2 


59.9 


o3 


9 


53-6 


09 


5 


72 


3 


53 


9 5- 6 


i3-4 


34-8 


60 • 1 


89.5 


23-5 


62-3 


06 


3 


56-i 


12 


2 


7 5 


1 


54 


97-5 


i5-4 


3b- 9 


62-2 


91-7 


25-8 


64-6 


08 


8 


58-7 


14 


8 


77 


9 


55 


99.4 


17-4 


39-0 


64.4 


93.9 


28-0 


67-0 


11 


2 


61 .2 


17 


5 


80 


6 


56 


44oi-4 


19.4 


4i* 


66-5 


96-1 


3o-3 


69.4 


i3 


7 


63-8 


20 


2 


83 


4 


57 


o3-3: 21-4 


43-0 


68-6 


98-3 


32-6 


71.7 


16 


1 


66-3 


22 


9 


86 


2 


58 


o5-3 23-4 


45-1 


70.7 


4900 • 5 


34-9 


74-1 


18 


6 


68-9 


25 


5 


89 





59 


07-2 25-4 


47-2 


72-9 


02'7 


3 7 .i 


76.4 


21. 1 


7i-4 


28-2 


91.8 



m 



MERIDIONAL PARTS. 



LATITUDE. 


MlN. 


69° 


70° 


71° 


72° 


73° 


74° 


75° 


76° 


77° 


78° 


79° 


Mix. 





5794.6 


5965.9 


6145.7 
48.8 


6334.8 


6534.4 


6745.7 


6970.3 


7210.1 


7467.2 


7744.6 


8045.7 





1 


97-4 


68.8 


38.i 


37.8 


49.4 


74.2 


14.2 


71.7 


49.4 


5i.o 


1 


2 


58oo. 1 


71.8 


51.9 


4i.3 


4i.3 


53.o 


78.1 


i8.3 


76.1 
80.6 


54.2 


56.2 


2 


3 


02.9 


74-7 


54.9 


44-6 


44.7 


56.6 


81.9 
85.8 


22.5 


59.0 


6-1.5 


3 


4 


o5.7 


77-6 


58.o 


47-8 


48.1 


6o.3 


26.6 


85.o 


63. 9 


66.7 


4 


5 


o8.5 


80.6 


61.1 


5i.i 


5i.6 


63. 9 


89.7 


3o.8 


89.5 


68.7 
73.5 


72.0 


5 


6 


II.3 


83.5 


64.2 


54.3 


55.o 


67.6 


93.6 


35.o 


94.0 


77-3 
82.6 


6 


7 


14.2 


86.4 


67.3 


57.6 


58.5 


jjr.fr 


97.5 


39.1 


98.5 


78.4 


7 


8 


17.0 


89.4 


70.4 


60.8 


61.9 


74-9 

78.5 


7001.4 


43.3 


75o3.o 


83.3 


87.9 


8 


9 


19.8 


92.3 


73.5 


64.1 


65.3 


o5.3 


47-5 


07.4 


88.1 


93.2 


9 


10 


5822.6 


5995.3 


6176.6 


6367.4 


6568.8 


6782.2 


7009.2 
i3.i 


725.1.7 


75 1 1.9 
16.5 


7793.0 


8098.5 


10 


11 


25.4 


98.2 


79-7 


70.6 


72.3 


85.9 


55.8 


„97-9 
7802.8 


8io3.8 


11 


12 


28.2 


6001.2 


82.8 


73.9 


75.7 


89.6 


17.0 


60.0 


fti.o 


09.2 


12 


13 


3i.o 


04.1 


85. 9 


77.2 
80.4 


79.2 


93.2 


20.9 
24.8 


64.2 


25.5 


07.7 


14.5 


13 


14 


33.8 


07.1 


89.0 


82.6 


96.9 
6800.6 


68.4 


3o.o 


1-2.6 


19.9 

25.2 


14 


15 


36. 7 


10.0 


92.1 


83.7 


86.1 


28.8 


72.6 


34.5 


17.5 


15 


16 


3 9 .5 


i3.o 


95.2 


87.0 


89.6 


04.3 


32. 7 


76.8 


39.1 
43.6 


22.4 


3o.6 


16 


17 


42.3 


16.0 


98.3 


90.3 


93.0 


08.0 


36.6 


Si. 1 


27J 


36.o 


17 


18 


45.1 


18.9 


6201.4 


9 3.6 


96.5 


41.6 


40.6 


85.3 


48.1 


32.2 


4i.3 


18 


19 
20 


48.0 


21.9 


o4.5 


969 


6600.0 


i5.4 


44-5 


89.5 


52.7 


3 7 .2 


46.7 


19 


585o.8 


6024.9 
27.8 


6207.7 


6400.2 


66o3.5 


6819.1 


7048.5 


7293.7 


7557.3 


7842.I 


8i52.i 


20 


21 


53.6 


10.8 


o3.4 


07.0 


22.8 


52.4 


98.0 


61.8 


47-1 


5 7 .5 


21 


22 


56.5 


3o.8 


13.9 


06.7 


io.5 


26.5 


56.4 


73o2.2 


66.4 


52.0 


62.9 


22 


23 


5 9 .3 


33.8 


17.0 


10.1 


1 40 


3o.2 


6o,3 


06.4 


71.0 


57.0 


68.4 


23 


24 


62.2 


36.8 


20.2 


i3.4 


17.5 


33.9 


64.3 


10.7 


75.5 
80.1 


61.9 


73.8 


24 


25 


65.o 


3 9 .8 


23.3 


16.7 


21.Q 


37.6 


68.3 


i5.o 


66.9 


79.2 
84.7 


25 


26 


67.8 


42.7 


26.5 


20.0 


24.5 


41.3 


72.2 


19.2 
23.5 


84.7 


71.9 


26 


27 


70.7 


45.7 


29.6 


23.3 


28.0 


45.1 


76,2 


89.3 


76.9 


90.1 


27 


28 


73.5 


48.7 


32. 7 


26.6 


3 1. 5 


48.8 


80.2 


27.7 


lit 


81.9 


9 5.6 


28 


29 


764 


5i. 7 


35.9 


29.9 


35.o 


52.5 


84.2 


32.0 


86.9 


8201. 1 


29 


30 


58 79 . 2 
82.1 


6054.7 


6239.0 


6433.3 


6638.5 


6856.3 


7088.2* 


7 336.3 


7603.2 


7891.9 


8206.6 


30 


31 


57.7 


42.2 


36.6 


42.1 


60.0 


92.2 


40.6 


07.8 


96.9 


12. 1 


31 


32 


85.o 


60.7 


45.3 


3o. 9 
4I2 


45.6 


63.8 


96.2 


44-9 


12.4 


7902.0 


17.6 


32 


33 


87.8 


63.7 


48.5 


49- 1 


67.5 


7100.2 


49.2 


17.0 


07.0 


23.1 


88 


34 


90.7 


66.7 


5i. 7 


46.6 


52.6 


71.3 


04.2 


53.5 


21.7 


12.0 


28.6 


34 


85 


9 3.6 


69.7 


54.8 


49.9 
53.3 


56.2 


75.0 


08.2 


5 7 .8 


26.3 


I7.I 


34.1 


85 


36 


96.4 


72.7 


58.o 


59.7 
63.3 


78.8 


12.2 


62.1 


3i.o 


22.1 


3o. 7 
45.2 


36 


37 


99.3 


75.7 


61.2 


56.6 


82.6 


i6.3 


66.4 


35.6 


27.2 


87 


38 


5902.2 


78.8 
81.8 


64.3 


60.0 


66.8 


86.3 


20.3 


70.7 


4o.3 


32.3 


5o.8 


88 


39 
40 


o5.o 


67.5 


63.3 


70.4 


90.1 


24.3 


7 5., 


45.o 


37.3 


56.3 


39 


5907.0 
10.8 


6084.8 


6270.7 


6466.7 


6673.9 


6893.9 


7128.4 


7379.4 
83.7 


7649.7 


7942.4 


8261.9 
67.5 


40 


41 


87.8 


73.9 


70.0 


77-4 
81.0 


6901.5 


32.4 


54.3 


47.5 


41 


42 


i3. 7 


90.8 


77-i 


73.4 


36.4 


88.1 


59.0 


52.6 


7 3.i 


42 


48 


16.6 


93.9 


802 


76.7 


84.6 


o5.3 


40.5 


92.4 


63.7 


57.7 


78.6 


43 


44 


19-4 


96.9 


83.4 


80.1 


88.2 


09.1 


44.5 


96.8 


68.4 


62.8 


84.3 


44 


45 


22.3 


99.9 


86.6 


83.5 


91.7 


12.9 


48.6 


7401. 1 


7 3.2 


68.0 


95.5 


45 


46 


25.2 


6io3.o 


89.8 
93.0 


86.9 


9 5.3 


16.7 


&2.7 


o5.5 


77*9 
82.6 


7 3.i 


46 


47 


28.1 


06.0 


90.2 


98.9 
6702.5 


20.5 


56.7 


09.9 


78.2 


83oi.i 


47 


48 


3i.o 


09.0 


96.? 


9 3.6 


24.3 


60.8 


14-3 


87.3 


83.4 


06.7 


48 


49 


33.9 


12. 1 


99.4 


97.0 


06.1 


28.1 


64.9 


18.6 


92.1 


88.5 


12.4 


49 


50 


5936.8 


6n5.i 


63o2.6 


65oo.4 


6709.7 

l3.2 


693 1 .9 


7169.0 


7423.0 


7696.8 


7993.7 


83 1 8. 1 


50 


51 


39.7 


18.2 


o5.8 


o3.8 


35.7 


7I1 


27.4 


7701.5 


98.9 


23.8 


51 


52 


42.6 


21.2 


00. 1 


07.2 


16.8 


3 9 .6 
43.4 


77.2 


3i.8 


o6.3 


8004.0 


204 
35.1 


52 


53 


45.5 


24.3 


12.3 


10.6 


20.4 


81.2 


36.2 


11. 1 


09.2 


53 


54 


48.4 


27.3 


i5.5 


14.0 


24.0 


47-2 


85.3 


40.6 


i5.8 


14.4 


40.8 


54 


55 


5i.3 


3o.4 


18.7 


17-4 


27.7 


5i.i 


89.5 
9 3.6 


45.1 


20.6 


19.6 


46.5 


55 


56 


54.2 


33.4 


21 9 


20.8 


3i.3 


54-9 
58.8 


49-5 


25.4 


24.8 


52.2 


56 


57 


57.2 


36.5 


25.1 


24.2 


34-9 


97-7 


53.9 
58.3 


3o.2 


3o.o 


58.0 


57 


58 


60.1 


3 9 .6 


284 


27.6 


38.5 


62.6 


7201.8 


35.o 


35.2 


63. 7 j 


58 


59 


63.o 


42.6 


3i.6 


3i.o 


42.1 


46.5 


05.9 


6J.8 


39.8 


4o.5 


69.4J 


59 



